Master Thesis Class Field Theory Artin

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. uevsr nv-rf il-n.Dr. Dipl.-Ing. Univ.-Prof. Supervisor: nttt fDsrt ahmtc and Mathematics Discrete of Institute inaUiest fTechnology of University Vienna uhr oksPapadogiannis Loukas Author: RI EIRCT LAW RECIPROCITY ARTIN LS IL THEORY FIELD CLASS ATRTHESIS MASTER ihe Drmota Michael pi 2018 April Geometry Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oeGli hoy,fial ioosapoc fpont rusi ie by given in is (for groups Als text profinite our theory. of approach for field rigorous class prerequisite [2]. a global a Neukirch finally of are theory), section Algebra Galois the w basic some that in of out approach find structures his may some extend reader the great literatu actually a the text, to to excellent easily an very is refer which may [1], reader the since meaningless I Vienna). studies. interesti (U very Mahnkopf Mathematics this Joachim in to Prof master by introduction my my area mention cal of to Drmot completion like Michael towards would Prof work also of a supervision as the under enna) written was thesis This Acknowledgements em fiel)i o eesr o h olwn chapters. following the the (Main for 5 necessary chapter not parenthetical is the any ideals) Also nowadays L-serie of used topic. terms to algebraic not refering shoul purely are one a chapter methods in course The analytical Of but theory. work. approach, valuation theory, this interesting glob of on cohomology of body chapter understanding of the main language an by the followed the constitute theory on theory, field background field class abstract a of gained chapters have the we as soon h hpeso h etmyb edsqetal fcus,btstil but course, of sequentially read be may text the of chapters The be would it theory, cohomology to introduction an give to not prefered We 2 gmathemati- ng e eNeukirch ie re, T Vi- (TU a followed e rm in orems lclass al stance, san is s have d more as l o Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. nta edt ofotsc eet si a considered. analytic was an it as of defect, exception a pion such the the confront of with effort to much theor field algebraic, after field that nowadays class purely in redundant mostly local is is of class which use idele case approach how using we the see theory in that field can groups class language we multiplicative fram global and or of the theory case modules give We the concentrate as field in will class case. translated basic Abstract notes be the can introducing these Apar is theory cases, which itself. the field general extensions, of the abelian more of of arithmetic the case the prov about the of Theory, terms remarks Galoi in the Field few field describe Class to global a is or in theory local field introduction a class of of an goal The give law. we reciprocity Thesis this In Abstract extensions s n Artin ing .The y. groups from t ework eers this on Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. btatcasfil theory field class Abstract 4 h hoyo valuations of theory The 3 L-series 2 Introduction 1 Contents anTerm ntrso das39 theory field class Global 6 ideals of terms in Theorems Main 5 . ento fpai ubr 22 ...... The numbers p-adic of 3.2 Definition 3.1 . xesoso autos...... 26 ...... valuations of Extensions 3.3 . ri a 15 . 19 ...... 8 ...... 12 ...... Theory . . Field . . Class . . Abelian . . Non ...... 2.4 ...... map inequality . Artin index . Norm second . 2.3 The . methods analytic 2.2 Some 2.1 . ls edter 4 5 ...... theory field . Class . of Aim . theory field 1.2 Class 1.1 . ento fAtno eirct a 39 . 29 ...... map . reciprocity . or Artin . of . Definition . . 5.1 . . formation class of Definition 4.1 . h eirct a 70 64 ...... 54 . . 58 ...... 44 . 47 ...... law . . reciprocity . . . . . The . . . . . 53 group . invariants . . 6.7 Class Idele . . Idele . the . classes . of idele 6.6 . group Cohomology . and idele . ideles the . on . of 6.5 . based Cohomology . formulation . the . . of 6.4 . Continuation . . . 6.3 . group . class idele . Generalized classes Idele and 6.2 Ideles 6.1 .. est hoes...... 17 ...... 17 ...... extentions . infinite . and theory . field . class . Local . theorems 2.3.2 Density 2.3.1 .. h omrsdegop...... 50 ...... group residue norm The 6.2.1 p ai bouevle...... 24 ...... value absolute -adic 2 29 22 44 4 7 Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. elydigrsac,js rigt utvt myself. cultivate to trying just research, doing really aebe edn hvle’ e oko ls edter;Ia not am I theory; 1956 field Grothendieck, class on book new Chevalley’s reading been have I 3 Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. in,and tion), in he ie fiqiy h rtwsteqeto fslaiiyb r by solvability of question in the was numbers first algebraic The inquiry. of lines three aldaein(ylc,if (cyclic), abelian called ento 1.1.2. Definition etrsaeover space vector trtdetnino bla xesossc as such extensions abelian of extension iterated some frtoa numbers rational of hoe 1.1.3 Theorem field oiainta e oteiiito fcasfil hoy[] n h scop the and [7], 1.1.1. theory Definition field class of introd initiation short thesis. the a this to giving led before that definitions, following motivation the give theory first We field Class 1.1 Introduction 1 Chapter ovd huhoecndemta utn dersac scmn closer. coming is research of h edge extensions problem cutting abelian that this ever, dream abelian Unfortunately, can understanding one to fields. though radicals solved, number of of terms r extensions in identifyi its cyclic) written of above problem be subextension the can each reduces numbers of criterion group This Galois inclusion. the tive written have we where ru ftesltigfil of field splitting the of group l euvlnl,sm)rosof roots some) (equivalently, all n aossoe hta reuil polynomial irreducible an that showed Galois and Q hti,i h pitn edof field splitting the if is, That ls il hoy(F)eegdi h ieenhcnuyfo at from century nineteenth the in emerged (CFT) theory Field Class ( K ζ n n a ot htcnb xrse i aiasi n nyi h Galois the if only and if radicals via expressed be can that roots has , sabifrmne,[ reminder, brief a As . ) o oen where n, some for Gal ( Q (Kronecker-Weber) ( Q ζ n ubrfield number A . Let Q ) / eremastedmnino h edcniee sa as considered field the of dimension the means Degree . Q Q Q L/K ( = ) ol eepesdusing expressed be could L/K ⊆ Q Z ζ / n Z r known: are f eetnino ubrfils hsetninis extension This fields. number of extension be 2 sGli n Gal( and Galois is Z /n saprimitive a is Q sslal,ta s h pitn edof field splitting the is, that solvable, is f Q Z ( a ewitna ainlfntosof functions rational as written be can ( K ζ ) f ζ n ∗ . 3 : ) iha element an with , safiiedge edetnino h field the of extension field degree finite a is ∈ ) vr bla xeso of extension abelian Every ⊆ 4 Q Q Z [ / = ] x 3 a naeinGli ru,then group, Galois abelian an has ] Z Q n φ hro funity. of root th ( ( ζ n 3 L/K f i,teElrttetfunc- totient Euler the (ie, ) , n ( √ 3 hros us t. Abel etc.? sums, roots, th x 2) ) saein(cyclic). abelian is ) ∈ m K ∈ [ x gwihalgebraic which ng ( o oenumber some for ] Z /n Q dcl:which adicals: Z cino the of uction ) sntbeen not as scontained is ∗ cigas acting o even (or f espec- ζ How- san is n least of e for Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. a ouini n nyi tde in does it if only and if solution a has naqartcextension quadratic a in hoe 1.1.4 theorem: typical Theorem a of example an is ing ieyagrtmcly ecnrcs uhpolm sakn if asking as problems such be recast can can problem We this that algorithmically. out tirely turns it properties; congruence ementary for rfr opeettig ndaetctrs o edcont field a how terms, transcending. dialectic own in its things present to prefers ubrfield number Q (where rtmtccneune.Frisac,iette like identities instance, For consequences. arithmetic ζ ubrfields. number h i fcasfil hoyi h following the is theory theory field field class of Class aim The of Aim 1.2 reappearance a make will global, and local connecting of idea broader the and r rdce yteKoekrWbrterm(ic hs numb these (since group theorem Galois Kronecker-Weber abelian the associated by predicted are ewl e,Gusepandta o-biu dniisin identities non-obvious that explained Gauss see, will we htw hudatmtt nesadsc dniismr fully. more identities such understand to attempt should we that au in datum n u ie diinl(huhiepii)cnrlo h iuto o gen for situation the of control inexplicit) (though additional gives but , hvle 1940 Chevalley 1 7→ hcigfrsltosover solutions for Checking h eodqeto a hto nigiette o leri number algebraic for identities finding of that was question second The Let • ial,tetidae a ovblt fDohnieeutos h f The equations. Diophantine of solvability was area third the Finally, • h beto ls edter st hwhwteaeinexte abelian the how show to is theory field class of object The a i lrf l bla extensions abelian all clarify ecietesltigo rmsi naeinextension abelian an in primes of splitting the describe a , ζ ,y x, n K/ m ij √ F sesnilyeuvln oteKoekrWbrtermf theorem Kronecker-Weber the to equivalent essentially is CFT . K ∈ Q − K ∈ K = 3 hc satce to attached is which eanme ed then: field, number a be a edtrie yojcsdanfo u nweg of knowledge our from drawn objects by determined be can √ Q hnfrany for Then . ,wihi h ads aeo h bv nwy.Ti question, This anyways. above the of case hardest the is which ), = 2 ζ HsePrinciple) (Hasse 3 − ζ 8 ζ q + 3 − ( x 1 Q ζ 1 8 − = . x .., , ( √ 1 ζ = d 3 y ) n + R / ζ = ) q ∈ Z Q 8 ( ζ ses,adover and easy, is / + L x tlatfrtenorm the for least at , 3 K L . 2 1 . ( = X Z ζ . x .., , h equation the , Let of i 8 .Teeaihei osqecsindicate consequences arithmetic These ). 1+ (1 = R 5 − a K K i n + 1 n in and x = ) ydt hc r tahdto attached are which data by i 2 eanme ed and field, number a be + 1 √ : y i X − ) i 6= / 3) Q √ i Q a p / (1 + 2 iswti tefteeeet of elements the itself within ains ij p ( + 2 o l primes all for x h rbe eue oel- to reduces problem the i x N j − − 1 ( x i − soso nalgebraic an of nsions ) Q + / √ √ y aenon-trivial have y L √ − 2 K ∈ d 3) of p tef ri one if or itself; = ) Q r aean have ers . / 2 K ovden- solved sanorm a is x 2 ythe by ollow- K − .As s. eral . dy or 2 . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. h lmnsaeuisa l places all at units are elements the h ideal the en h ru fpicpliel”rm to ”prime ideal principal of group the means K with nqaiisle ttecnr fCFT! of centre the at lies inequalities ehd.Frtefis nqaiyw ol aet s techniques use to have would we inequality first the For methods. hoyi hsetbihdasrcl n h iclywl et p the to from be transfered will be difficulty can our the formation” of ”class and framework of main abstractly properties introduc The established the we formation”. thus where ”class is theory, define theory and field group” class ”profinite abstract of to further proceed theory.. cohomology on absolutely based proach rmlilctc ru ntecs flclcasfil hoy esatw start th We field theory. field class class global local of of group case class the idele in the group multiplicatice to or theory, field class abstract hc satce to attached is which nthese, In ttecnr foraayi r the are analysis our of centre the At • ewl kpta eodse n rce ute nta,t anothe to instead, further proceed and step second that skip will we • • analyti using inequality • index Norm second the prove to is goal first Our • Let • oad hsts esalgv oeitouto ovlainter and theory valuation to introduction some give shall we task this Towards ecieteGli ru fa bla extension abelian an of group Galois the describe )[ 2) ubrter fetnin flclfields... local of extensions of theory number cohomology theory group number (some) of foundations theoretic Ideles )[ 1) P K L/K ( J J m m K K .I hpe ,w ieterlvn eal.Tepofo these of proof The details. relevant the give we 5, chapter In ). sdfie n5 in defined is ( ( J m m eacci xeso,then: extension, cyclic a be K : ) : ) ( m N N en h ru ffatoa da pieto ”prime ideal fractional of group the means ) L/K L/K J J L L L . hsotie by obtained this , ( ( 1 m m . .W a loietf norltranalysis later our in identify also can We 1. ) ) P P K K ( ( m m )] )] v 6 > dividing 6 [ [ L L ominequalities Norm : : K K ri eirct law reciprocity Artin m ] ] m and ” n r even are and ≡ L ( 1 of mod K : ytedtmin datum the by ≡ G m m -module ( 1 and ” h notion the e ”(elocally (ie )” from . o v mod oethat rove i P ( ith.. K K ap- r eory, A m m ( m v , 1 of ), c ) ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. eak2.0.4. Remark ento 2.0.1. Definition da if ideal ento 2.0.2. Definition field ento 2.0.3. Definition ag[]adavr ielcueta tedda h nvriyo Vien of university in the extracting at and attended 2 methods I theorem analytic i relevant that using the our [8], lecture framework following Mahnkopf nice inequality, Joachim very index Prof. a by Norm and second [6] the Lang now prove shall We L-series 2 Chapter osdrda eetutl14,wr ueyagbacpof eegiv were proofs algebraic purely Chevalley. were w by and 1940, nowadays effort until redundant much defect mostly are a form as in considered algebraic purely are that rems scle h iihe eisatce to attached series Dirichlet the called is iihe hrce flevel of character Dirichlet A p ab Let K 6= s ∈ . ∈ K/ R p C and ⇒ Q eanme ed and field, number a be a L ∈ L If : ( ( p ,s χ, ,s χ, Let Let : Let χ rb or χ : = ) = ) = m J R ∈ ( χ = id m L eacmuaiern.A ideal An ring. commutative a be L p eaDrcltcaatro level of character Dirichlet a be ) m 1 Q stetiilDrcltcaatro ee ,then 1, level of character Dirichlet trivial the is /P ( ( o ,b a, for d s id, ,s χ, p m p m m → = ) = ) . p 2 rmodulo or . ≤ S .Aayi ehd ob ie otheo- to given be to methods Analytic 4. O ∈ u 1 u O K ≤ 7 ≤ X = R O K O etern fitgr nornumber our in integers of ring the be K K { X ea ideal. an be , z ( χ 1 u ∈ /N ( , , m m C )=1 N ( u samrhs fgroups of morphism a is : = ) | χ s z ( | N =: u 1 = ) Q K /N Z ). } K ( ( u m s ) p ) s hnteseries the Then . ∈ R saprime a is nafter en esof deas (2.2) (2.1) this ere na Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ovre boueyaduiomyfor uniformly and absolutely converges . oeaayi methods 2.1.1. Lemma analytic Some 2.1 function. zeta Riemann the Proof. steDdkn eafunction zeta Dedekind the is ideals obtain ovre boueyaduiomyfor uniformly and absolutely converges p σ 1 − ovre boueyaduiomyin uniformly and absolutely converges ic o n rm number prime any for Since Let n ewn oso that show to want we and fi addition in If Remark: | 1 1 − p ewrite We 6 p ≤ N 1 ≤ ( ( O p p = O K ) − − 1 p K sth , Q s ∈ 1) , prime N,p k ∞ − Let epieielwith ideal prime be s σ =1 K X = 1 p | 1+ = L | δ p σ = ( eobtain we , ∈ Q ,s χ, + a p p | R eobtain we , ≤ k ≤ 1 X it p p O , O ovre absolutely converges = ) | − ≤ N w aet hwta h nnt series infinite the that show to have ,we p K > δ K O p | N , ( ≤ X , K p 1 X X p p Z L O primideal , ( primideal ) − p p K 0 p K Y − 1 primideal ( hnteifiieproduct infinite the then , ) ∈ ( s N | d s id, − s 1 1 ) N s ( − | p of p hr r tmost at are there ) = ) = N − | 8 s p 1 s K 1 1 s ( , | ( ∈ − ∈ N | − p . 1 − n { X ) p ∈ p C s {z − C N ( − N fs 1 N p 1 1 ∈ ∈ s | 1 ) ( satisfying N ( with 1 } s p 6 p − 1 C ⇔ N /n ) ( − ) − p − X n put and ) 1 : 1 ) s s P p s | − Re | Re =: − − s = k ∞ X p =1 ( ( (2.3) 1 | 1 s ζ p s | n ) ( Re ) p a s ( fs > [ = > p k ) ( 1 − ovreabsolutely. converge + 1 s − f + 1 1 K ) 1) 1 = > : | ,δ> δ δ, σ δ Q + 1 f 6 . 6 p ayprime many ] | p | p hnwe Then . δ fσ . 0 } 1 . − (2.4) 1 | 6 Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. p of Proof. hnw obtain we then and s rpsto 2.1.2. Proposition 1 hn( when .., , L b definition (by • h atsmconverges sum last The ovre boueyaduiomyi h set the in uniformly and absolutely converges e now Let hspoutlclzsoe rm das.adi h Elrprodu ”Euler the is and ideals.. prime over localizes product this • ( u ,s χ, )Frall For 2) )Let 1) p u r iiil nyb rm ideals prime by only divisible )Let 1) r nerliel,ntpieideals prime not ideals, integral are ≤ ) Re ”. O ( δ s s K ) ∈ ∈ L A h rm daswith ideals prime the > s C R ( N ,s χ, n + 1 ∈ χ , , with · , > δ C N p X N = ) ∈ p k δ Let p , 1 sth N ) ( ,..,k , k p ∈ r utpiaieadtesmrn xcl over exactly runs sum the and multiplicative are 0 ( σ ⇒ X 1 p , p ( hnteDrcltseries Dirichlet the Then . m ,..,k , χ p ≤ m = Re N r )=1 X ( O ∈ − 2 )=1 ⇒ Y u u eaDrcltcaatro level of character Dirichlet a be 1 Re r N . K Y ≤O X , ( etesto l ideals all of set the be ovre uniformly. converges 4 m ∈ ,N 1) s = 0 , ,N 2 N ( )=1 ) χ ( p ( K σ . 0 s u , p ovre boueyadde o eedon depend not does and absolutely converges 4 > ( ( m X ∈ ) p ) i Y =: 6 =1 | i Y 6 )=1 ) r A N =1 > 1 Y 6 r χ u N N {z n N ≤ ( χ ( ehave we + 1 N , p u k u χ X p O ( · p i k ∈ ) 1 primeideal ) p 9 ( K ( } k i s X N u i p ∈ − ≤ ) ) δ ) i = N N χ k ) N eobtain we , χ i ( O 6 N ( k L p ( ( 1 i Y K p σ u ) =1 m r ( 1 N k ) ) p ( N − 6 N ( sth i ,s χ, 1 p ) s n ( and ( ( { − n i k p − p s i k ) ) ) ) · N i − − ∈ − χ s k X s ks ∈ s ( ( u = p C p = ) p N = , 1 ) ) = ≤ m : N k 6 Re 1 = ) 1+ ( 1 O p m N δ ( ) K s − ednt by denote We . ) s t ( sth > + 1 texpansion ct u , δ m } 1 = ) A (2.5) N ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Proof. Q iihe series Dirichlet ovrefor converge yteaoeLmafor Lemma above the by ueyaduiomyfor uniformly and lutely P oain2.1.4. Notation 2.1.3. Corollary ( sth L s 2.1.7. Notation tl icl w ilsv h ieo on this). value doing the of study time can the we particular save In will (we difficult still )If 2) eoopi otnainto continuation meromorphic a hoe 2.1.5. Theorem eiu qa to equal Residue Then eak2.1.6. Remark ( nltccniuto osm range some to continuation Analytic ewl nyne xsec fa nltccniuto oaset a to continuation analytic an of existence need only will we ie > δ ( 1 = p u ,s χ, ) ≤ 6 · • nteaoeLmaadi at1 ehv enta ohsdso 2 of sides both that seen have we 1) part in and Lemma above the In hspoutcnegsasltl n nfrl in uniformly and absolutely converges product This · • P 2) |{z} ⇒ 2 O = O 0) f . hnw write we then , 5 u χ K L ) K claim. L nfr ii fhlmrhcfntosi gi holomorphic. again is functions holomorphic of limit uniform ∈ . s that Use and , ( scle h iihe -ucinatce to attached L-function Dirichlet the called is , Q ( A ( ( ,s χ, = p u ,s χ, N , , p m m p , | id ≤ ( )=1 g ) )=1 χ p = ) O , ( a ooopi otnainto continuation holomorphic a has aetesm rnia ati hi arn expansions. Laurent their in part principal same the have N m u K flevel of )=1 1 ) N , L N ( χ P | − p h ro of proof The : ∞ → ( J ( ( , ,N h ooopi ucinin function holomorphic The : ,s χ, ( )Let 1) m u If : u u ( Y χ u o n iihe character Dirichlet any For ) ) m )=1 ( ) s ( N p f − χ p ) ) ) /P 1 6 ) ( s m ( ,N nfc,telf adsd ovre biul to obviously converges side hand left the fact, In . ∽ ersnsahlmrhcfnto nteset the in function holomorphic a represents N ,g f, u | u N N ) ) = ( m N χ ( g s p then 1 p | ) . ∞ → P 6 ) odnt htteei ooopi function holomorphic a is there that denote to ovre nfrl nany in uniformly converges : r eoopi ucin nanihorodof neighbourhood a in functions meromorphic are − ∞ → − N J u s f N ( ∈ 1 Z m ( 2 n h ih adsd ovre biul to obviously converges side hand right the and A s ( s . − K . ) p N 1 = ) Hence . 1 /P ) ∈ ( . χ | − 5 s N ( L := ) m (1+ C Re p sln epaddifficult. and deep long is g ( 10 ( 1 ) ( u ,s χ, → N s δ ihol ipepl at pole simple a only with ( ) + ) ) s − ( ) L S p σ P ) > ) | 1 ( at ξ − ,s χ, u 6 ( eaDrcltcharacter, Dirichlet a be 1 ∈ s s s A P ) − = ) χ N s 1 = Re u δ ”eafnto of function (”Zeta u ∈ χ ∈ flevel of , ∈ ( A ( . C A ( χ u s > δ N N ) . X ) N , Re N { > u s ≤ ( ( 0) u 1 ( O ∈ u ) s m ) K − hc sdfie by defined is which ) − C smc airbut easier much is s > N (1+ thlsta the that holds it χ : ovre abso- converges ( Re Re ( + 1 u δ u ) ) ) ( s ( s s s δ Re ) ) ( , 1 = K > > ( χ > δ s )has ”) + 1 1 ) 6= (2.6) with − > ξ 0) ( id δ s δ . 1 6 } ) . , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Proof. for bouey hrfr emyrarnea follows as rearrange may we therefore absolutely, rpsto 2.1.8. Proposition Re ( lgigi h oe eiso o efrhrobtain further we log of series power the in plugging ovre at converges eseta h oaih series logarithm the that see we • eoti that obtain we For • ic the since • since > σ s ) )Teseries The 2) )teeaea most at are there 3) 1) > saayi nnihorodof neighbourhood in analytic is pliglgt ue rdc of product Euler to log Applying Re | 2 1 1 k 1 + | ( N χ s χ ( δ ) ( ( p ( , p p > ) log ) ) log N > δ sk k log z h sum the 1 ( k X L | = p L log | > P 6 ( ) L 0) ( 2 ,s χ, − χ ,s χ, ( k o l iihe characters Dirichlet all For ( s L ,s χ, k ( 1 > p {z | p ,m X ( 2 = ) p = ) ,s χ, = ) 1 N kσ k )=1 1 = ) N N ( p } ( ) log(1 p P 1 hr ( where − kσ p ( | [ = ) p , X p k 1 m − ≤ p p ) )=1 ≤ − s ≤ N ovre asltl)b h utetcriterion quotient the by (absolutely) converges O K χ O σ O K − ( ( X K K p p : = , X N ( ) ) , z p χ Q p ( ... sk k = ) | s=1 , p p ( ( m p aypieiel yn above lying ideals prime many ] , p p 11 − m ) | )=1 swl stesum the as well as ) ) σf )=1 6 s ∽ − + k X N 6 k X p L log(1 > > ≤ k X 1 ( · p > 1 O ,s χ, k − X k 1 2 K k 1 > σ ( ( Primideal, ,P z p 2 χ − eoti for obtain we ) 6 χ , X k m k ( 1 χ p )=1 flevel of 1 p ) X ( N 1 kσ p since ) k ( 1 N p < N ) ( χ ( − p ∞ p P ( ( ) s m , p p m − ) ) ( ) k k )=1 s sk > σ k > thlsthat holds it ) 1 N k 1 χ s 1) z ( ( k p p ∈ ) ) converge p s C ∈ (2.7) N with Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. eak2.2.2. Remark opimo groups of morphism ento 2.2.1. Definition osmlf oainw put we notation simplify to oko btatcasfil hoya edvlpi h etcatr,pr chapters, next the in develop t theory. following we cohomology so as do through theory to inequalities field avoid both class shall We grou abstract f number fields. class techniques of and local work use idele cohomology of to of group extensions have terms (some) of shall theory, theory in number we index of second inequaity analogous foundations the index pro theoretic the of norm shall to we first proof equal analysis, the the For is using theorem, ideals index main of this our terms that of in inequality statement inequality index the norm to index come Norm now We second The 2.2 em 2.2.3. Lemma Aim: • )Let 2) • in uniformly and absolutely converges series the hence Let hstescn umn n2 in summand second the Thus ⇒ )Drcltcaatr r hrceso h fiieaein group abelian) (finite the of characters are characters Dirichlet 1) 2) L/K G b claim. Prove χ sgopwrt group is ∈ xeso fnme ed with fields number of extension G b then , )Let 1) : Let : χ : [ g ( J G g X χ χ X K ∈ ∈ ∈ 1 G → G m ( G χ b G m N χ 2 χ = eafiieaeingop hrce of character A group. abelian finite a be S )( : ) ( then ( ( g 1 m g g p = ) edenote We . = ) P ≤ := ) = ) m O . K N shlmrhcin holomorphic is 7 Y ( ( ,prime N χ K L 0 | G 0 | 12 1 K G L J ( | L g J | else ) ( L fχ if else χ m fg if p ( m 2 m )] ( G Gal p b ) g 6 ) ∈ h e falcaatr of characters all of set the = 1 = [ ( J L L/K id K : K = ) ] Re ( Re G s ) n let and ( > s ) 2 1 > + 1 2 eltron later ve δ o ideles rom J + eframe- he K δ G ( m oving (2.8) G sa is ) /P ps. . m Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Then eassume We a write can 2.2.4. Theorem Proof. Proof. cycle the htfralcharacters all for that in L P where Let n o ecncm oorfinal our to come can we now and assume log obtain we log Applying ⇒ nteohrhn ehv eni h rpsto ftepeiu secti previous the of Proposition the in seen have we hand other the On umn vralcharacters all over Summing Then . g χ L P L ∈ eoi h rtpof ewl o edi. st h eodw have we second the to as it.. need not will we proof, first the omit we : eput We χ ( G nueacaatro h eeaie ls group class generalized the of character a induce ( ,s χ, g ,χ s, m ea rirr hrce fte(nt n bla)group abelian) and (finite the of character arbitrary an be ∈ χ g G u vrcosets over sum ( χ ( χ = ,χ s, ( = ) g k | = ) χ 6= = ) ∈ 6= ( Q J g |{z} g H shlmrhcat holomorphic is ) 7→ K id X id 0 = ) {z p m s ( gh ≤ hnteeis there Then . m := ec the hence , Let − χ O log ) P /H log( K 1) P } k g p ∈ L χ m L/K p m ∈ [ m ∈ J J ( N s G χ p K J p ,χ s, X K ∈ − K K L ( · usaanoe h hl group whole the over again runs k sdvsbeb l rm ideals prime all by divisible is ( m ( g rm ideals prime , J m m )+log( + 1) eayetnino ubrfilsadasm that assume and fields number of extension any be ) ) ( L /H ) ,χ s, : ) L ∼ , X ( X p m -series χ rm ideal prime P p χ | ) ) holomorphic ≤ ( h m k ≤ ∈ O N s ∈ ) K m g J p J χ | 1. = K {z L ,prime, G ∈ ( K L K χ 13 ( ,χ s, {z J \ k gh osaton constant ( X ( ( ,primeideal > L sth ,s χ, m m X ( } ) χ )) ) m ( 0 and ) } /H ( ( shlmrhcat holomorphic is ) χ )] p p − ∼ , ( ) eas ti holomorphic is it because m | χ h 6 N {z etu obtain thus we )=1 ( ) p ( h [ m ,w hnoti trivially obtain then we 1, 6= p L } ) k N ) =[ := χ N = − : χ ( · p s K p ( ( log ) p = p J = · − ] ) ) H K s s χ | J ( p s · 6=1 {z = ( N m K h − ≤ 1 : ) } ( ) ( p m 1 P O ) H ) − K s /P g s ∈ ] .Tu we Thus 1. = hc ramify which G = m ( χ χ J flevel of ( 6= K g ) ( ) id m ) ) /H on m .. . . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. n holds ing h N and χ ∈ · − log p J sn hsw obtain we this Using ie Thus ⇒ where • Then si rpsto npeiu eto adispof eseta h foll the that see we proof) its (and section previous in Proposition in As • obnn o ehv. nt htlog that (note have.. we now Combining ednt by denote we ∈ K aypieideals prime many X p \ H, ( p if H m ζ S sflydcmoe in decomposed fully is X p K ∈ ) p L/K /H prime otisalflydcmoe rm ideals. prime decomposed fully all contains ( ∈ p S s N L/K + ) ∈ k S ∈ ⊆ L/K N J (1 =[ := N K χ X H X K L ⇔ 6= ( ( = − m J p id S and log ) ) L L degk L/K /H χ k − log X ( ≤ 6= s m : O id 1 χ > k ) L L K X ( − k = m | 1 := ≤ ,degk k ( p h ,s χ, ≤ ,nt htoe any over that note ], ) χ s S then · H p log ) e L/K p m ∼ O ∈ k ∈ =1 ) k | L S p L ,primeideal log ∼ f L/K N . X N X n n such any and s k h e falpieideals prime all of set the | K ( − p ζ L 1 p ( L ⊆ ( o all for 1 = ) 1 ( k − H s )(= s ∼ ) ) 14 N + ∼ N h ( p ooopi nRe in holomorphic k · ( p | k f ≤ p p ) ≤ k ) ∈ − O | O p − H, s L k = ) L X k s X kprime ,k = ,degk p | a degree has ζ = p = p K X primeideal N p h ( ( h ∈ {z s > h 0 ) k · 2 ≤ S p ∼ N p | N O ∈ L/K − k k H, ( L fs ( s ( k {z p ,degk p s = 1 = = 1 6= N p − ,f ) ) 1 ) degk ≤ − − > rm ideal prime > X ( X hr r precisely are there 1 s } s p } 2 O =1 1 2 ) ) − K 1. = , H H s ( k, t ( sth ( m ∈ )=1 J N p K , ( N ( m p m (2.10) ) ( 1 = ) (2.9) − k ) ow- /H ) s − s ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ie lohdoeo h otbatfliesi ahmtc o ge for mathematics in Jugendtraum) ideas liebster Kronecker beautiful (the most fields the number of one had also lblfil ntrso h rtmtco h edisl.Fraeinex abelian For Kronecker itself. by field 1930 loc and a the 1850 of of roughly extensions between arithmetic Galois developed the the was theory of describe the to terms sions is in theory field field global class of goal The map Artin analytic These Takagi. 2.3 by theory field ideles. class with working of when theorems redundant are main methods the of proof m egbrodof neighborhood χ 1 oehrwt 2 with Together since ⇒ h da-hoei esoso hs nqaiiswr led us already were inequalities these of versions ideal-theoretic The obtain we Thus enwlet now we obtain we fw lgti neuto 2 equation in this plug we if h KRONECKER o all for 0 = · (1 (1 p ∈ H, − − N h p χ χ rm ideal prime > 6= 6= X id,χ id,χ srcl igr and bigger) (strictly 0 s χ X X → 6= s ∈ ∈ J J hsbuddadw ute obtain further we and bounded thus 1 = 12–81.H eeoe natraiet eeidside Dedekind’s to alternative an developed He (1823–1891). id . ( ( \ \ + hslog( thus 1+, hsyields this 9 m m ) ) N n hsw find we thus and /H /H [ (1 k L ( ≤ p m m − : O ) − χ χ K L X χ log ) ) s X ,degk 6= ] > > 1 id > . > 0w obtain we 10 log( |{z} m s N =1 [ h > s N J h − g χ 0 N − K h 1 · 2 N ) s 1) ( ( log 1 > s − m ( 15 rN or ) k > −∞ → : ) 1) ) N m s h − N − χ P s h 1 → − m ∼ > 1 log > N log( 0 + but , log o all for 0 K h L g s ooopi at holomorphic J 2 s − s ( L 1 s − ( − 1 g ) 1 m 2 1) eaigaeinetnin of extensions abelian nerating 1 ( + )] g | χ s 1 {z shlmrhci the in holomorphic is ) ooopi at holomorphic ( 6= s } ) id hsipisthat implies this , g | s=1 2 {z ( s } ) s=1 di the in ed l.He als. lor al ten- 1 , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. lsicto fcnrlsml lersoe ubrfields numb number over over algebras forms simple quadratic central of all classification for principle (local-global) scnetrdb ee n Hilbert and Weber by conjectured as fcasfil hoy(mrvmn fTkg’ eut) Int results). Takagi’s of (improvement theory field class of eeo ubrter,adhv lobe influential theory been the also of have account and theory, systematic number first on the were gave which 1897, in xeddt ihrdmninlfils hogotb nextension succesf an been by has much Throughout theory been th fields. field in has dimensional class less higher there abelian but to years 1980, cases, extended recent about field Beginning In function and case. t 1967. local field should in nonabelian theory Weil the the in to shape progress Langlands the from of letter indication a first the extensions abelian containing details. aicto.Md motn rgesi ls edter a theory field class in progress important Made ramification. Weber n where e field ber and ieyt h rm dasof ideals prime the to tively hoe 2.3.1. Theorem Let 19 When (completely). aiyin ramify p ojgc ls ( class conjugacy primes all of set the in K ojgc class conjugacy sffiin o 2 for (sufficient lse feuvln autosof valuations equivalent of classes n h ojcuewspoe yCeoae n1926. in Chebotarev by proved was conjecture the and 2 order . 3 soitdwt h oacieenvlain of valuations nonarchimedean the with associated = th 6 5 4 3 2 naie in unramified . e’ ealta prime a that recall Let’s Note: If HASSE ARTIN TAKAGI HILBERT WEBER .Foeiswsol bet rv ekrsaeetta i conjectu his than statement weaker a prove to able only was Frobenius 1. Spl e e etr rbnu rvdtefloigstatement: following the proved Frobenius century 1 i K f 2 f F L/K h aicto ne.When index. ramification the ( Hilbert , ( 1 i p L/K safiieagbacnme ed ema yteprimes the by mean we field, number algebraic finite a is K + nGads h ttmn ple otetiilcnuaycasgives class conjugacy trivial the to applied statement the so and G in ) r h rm dasof ideals prime the are L o n aosextension Galois any For K .. 19–99.H aetefis ro flclcasfil theor field class local of proof first the gave He (1898–1979). 19–92.H on h Atnrcpoiylw,wihis which law”, reciprocity “Artin the found He (1898–1962). when , ema that mean we sGli the Galois is 14–93.H on h orc eeaiaino “class of generalization correct the found He (1842–1913). 17–90.H rvdtefnaetlterm fabelian of theorems fundamental the proved He (1875–1960). + etesto rmsof primes of set the be ) o hsadtenx eto ecnrfrt in 4 o more for [4] Milne to refer can we section next the and this For . 16–93.H rt eyifleta oko leri nu algebraic on book influential very a wrote He (1862–1943). . e 3 3 C g L p Takagi , . f ) nwihcrancnuaycassaegopdtogether, grouped are classes conjugacy certain which in 1), L/K , when g n ojcue httedniyo h rmsgvn fixed a giving primes the of density the that conjectured and e is , i n = | C [ = feeet in elements of ) f | L/K / 4 i e | Artin , L p L G i 1 = O ′ | scnandi oefie lerial lsdfield closed algebraically fixed some in contained is s : p ycntuto h lmnsi ( in elements the construction By . L K K sGli,teset the Galois, is r qa osay to equal are of ( = ∀ ], p O i w s h aesymbol same the use (we 5 O O f : Hasse , L F L K i K p L/K 1 uhthat such [ = .. = hr edsigihbtentefinite the between distinguish we where , atr na extension an in factors = F K e 16 F i O g G F ) 1 e > pitn in splitting 6 L fnme ed,Foeisatce a attached Frobenius fields, number of 1 e 1 = n , .. .. / n tes erfrt 4.Frnon- For [4]. to refer we others, and o some for 1 F F F Gal i n g e F g : = e h prime the Spl i O ( ∩ and efg L/K K oue h ri L-series Artin the roduced O rfils n otiue othe to contributed and fields, er ( / L/K dteKoekrJugendtraum Kronecker the nd K p oeo i aosproblems famous his of Some . L f K aldteietadegree inertia the called ] oec rm ideal prime each to ) oad h n fthe of end the Towards . i = i htcrepn bijec- correspond that , ′ h prime the , s ) p p osay to a density has and p ssi osltin split to said is ,adteinfinite the and ), L ,poe h Hasse the proved y, ru”t lo for allow to group” of e h antheorem main the i f ls edtheory, field class ≥ p K and , L/K , L p p moreover 1 brtheory mber as of fanum- a of number e ssi to said is 1 k sin is ake / [ K have ) L the , : ully p K re of L ] Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. h elpie orsodbjcieyt h ieetembeddi different the to ones bijectively complex correspond and primes real real the The between further distinguish primes,that into n ( and ewn o opoe2 prove to now want We theorems Density 2.3.1 respectively). n h ope rmscrepn ietvl otepiso comple of pairs the to bijectively of correspond embeddings gate primes complex the and lal hr are there clearly ewn oitrrtti ntrso aosgop.Let group groups. Galois Galois with of fields terms number in of this tention sequences. interpret these to among want equidistributed We are numbers prime the hr xssaunique a exists there ti eoe s( as denoted is it field case n ( and sw etoe rbnu ojcue htfreeycnuayclass conjugacy every for that conjectured Frobenius mentioned we As 1 { lsie.Hwvrteewr set fteter htwr consid were that bee had theory fields the local of and aspects fields were number extentions there both However infinite of extentions and classified. abelian theory the field 1930 By class Local 2.3.2 G ih[]atn as acting [n] with iil y4 ehv h result the have We 4. by visible ehv ( have we ( p, /ϕ ( h est of density the , F Q a) b) e now Let o consider Now fGal( If o let Now L/K , ( C m [ σ O τ ,m a, ζ σ F ∀ m ntesto rm numbers. prime of set the in ) K rdc h aevlainof valuation same the produce suiul eemndb hs conditions. these by determined uniquely is α L/K , ] ∈ / / : ) F ,cnie o h sequence the now consider 1, = ) p Q ∈ L/K G L/K , , [ = ) ( τ O τ p F F F = ) L ie ) | = sntaeinw eoeb ( by denote we abelian not is ) p [ = eodpiedividing prime second a , p } ( = ) F ϕ ] τ { Q σα ζ F p ( ( α ∩ L/K , [ 7→ m F K ζ | ] σ K L/K , m ..a ( ≡ σ F = ) ( p ∈ ζ F into ] ′ L/K , ∈ / L/K , n α = ) − Q . prime A . ) Gal 3 q G | ,a a a, m, . ( mod ) ≡ o yltmcetnin of extentions cyclotomic for 1 , aldthe called , C Z τ F ( m /m − = ) eosreta w ojgt mednsof embeddings conjugate two that observe we , ) Q σ 1 when , [ h ro ses..Tu fGal( if Thus easy... is proof the , ihrodinteger odd either ∀ F ζ Z Gal C m F where , + ) , ∗ ] } p / F | ,a m, ( Q is ′ itntsqecs iihe hwdthat showed Dirichlet sequences. distinct Q o dividing not K | 17 te ( ,then ) [ F | rbnu element Frobenius p ζ C p 2 + w write (we m ewietu ( thus write we , surmfidover unramified is , | q / ] nme feeet nteresidue the in elements of =number τ / | ,.,a .., m, G G Q ∈ p o vr rm ideal prime every For . | ) . L/K , G = ∼ p, then , Q ( m p > Z + [ |∞ ζ /m ,tecnuaycasin class conjugacy the ), surmfidin unramified is rapstv nee di- integer positive a or 1 m km.. , ] Z / p G Q ) p ∤ ∗ ( Q L/K , [ = ) τ L/K ∞ Let . F p sth = ) eas nthis in because , o nnt,finite infinite, for g of ngs L/K ). τ eafiieex- finite a be τG a density has ] m . sabelian is ) ( > Q K F F rdun- ered [ conju- x ) ζ ( 2 τ into m in − (2.11) and ] ∈ C 1 O Z R K L in G n ) , , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. h oa ri asadsosthat shows and maps Artin local the httehomomorphism the that hndfie lblAtnmap Artin global a defines then sa pnsubgroup, open an is omttv diagram commutative h ignlsubgroup. diagonal the eto h group the of nent by induced lotalnnrhmda primes. nonarchimedean all almost les stobgfrteeeita exist there for big too is o opc rus n h groups the and groups, compact for u l h ale rosue nlss hvle aeapu a formula gave natural Chevalley a give extensions analysis; to abelian used able was proofs id`eles he of earlier introduction the all but aifcoy hc hvle ihhsnwnto fa dl a beto able was idele an of notion new Let his with Chevalley which satisfactory, ttmn flclcasfil hoyi htfrevery for that is theory field class local of statement eldfie homomorphism defined well a ri a hs opnnsaetelclAtnmp,i h es htthe that sense the in maps, primes Artin all a local for define the to commutes are possible diagram be following components might it whose that map suggests This Artin groups. norm are index ri map Artin o vr nt bla extention abelian finite every for 7 I nteChevalley the In hnedwdwt h oooyfrwhich for topology the with endowed When n rbe sthat is problem One CHEVALLEY K K ab ob h ugopof subgroup the be to eoetecmoieo l nt bla xetosof extentions abelian finite all of composite the denote ϕ htidcsteisomorphism the induces that ) ssretv ihkre qa oteiett once compo- connected identity the to equal kernel with surjective is 10–4.Temi ttmnso ls edter r pure are theory field class of statements main The (1909–84). I K 7 /i I prah n rvsfis oa ls edter directly, theory field class local first proves one approach, K ( K Q eoe oal opc ru.Embed group. compact locally a becomes Q K ∗ φ p .I te od h homomorphism the words other In ). K K hvle ovdti ydfiigtegopo ide- of group the defining by this solved Chevalley . I I p ∗ K K K /Nm Q   K p p ∗ ∗ p ∗ /K p p p ϕ ∗ Y ϕ sntlclycmat hswudb reonly true be would this compact, locally not is |∞ K K L/K ∗ : ( p ∗ L K → φ K K φ : φ ∗ K t tcnit ffmle ( families of consists it sth φ p / p I ∗ / p Gal ) ∗ ∗ p K K ?? Gal 18 Gal oevral(pn ugop ffinite of subgroups (open) all moreover ; × / −→ / r nylclycmat nfact In compact. locally only are Gal ∼ → Gal ∗ → p Y ( ( ∤ scnandi h enlof kernel the in contained is ( K ∞ K Gal K Gal ( Gal p ( K p ab  K O ab ab of p ab /K  ( p ∗ ab /K ( /K ( L/K K K K /K /K p ino ls edter o infinite for theory field class of tion ab ) ab ) ) p /K ) /K ) ) p eyagbacpof ihhis With proof. algebraic rely ai field -adic nwcle the called (now ) hs opnnsare components whose ) a K p ), ϕ hr exists there , K K K a h full The . p yalgebraic, ly ∗ t noa into fits ∈ in ϕ Q O K global I solve. local p p ∗ K and K for as p ∗ Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. iediagram tive bla extension abelian cie stelclcmoeto h lblAtnmp u twsntuntil not was it. 2.3.2. it of Notation but description de local map, be explicit Artin could an global it found Tate the or and locally, of Lubin characterized component that be local 1965 could the map as Artin scribed local the 1950 By hudtk,o vnwehri xse.I 96 ri pcltdta fi theory that speculated field Artin class 1946, nonabelian In a existed. form it whether what even unclear or was take, su it should abelian decades largest its several and For field class extension (ray) an the sion. between of distinguish subgroups fie of the not function that do part the shows theorem and is limitation case norm theory local The field the in class especially conjectures [4]. abelian made, of ques Non series been interlocking the main has vast the raised progress theory.. a Takagi of is field world, which proof program, the his class Langland’s announced to nonabelian he theory which a field in of class 1920 abelian ICM Theory the of at theorems Field talk same Class the In Abelian Non 2.4 t used He sequence. exact theory. to long field restriction, single class and of transfer a group of reformulation finite in properties a with combined for together suitably groups sequence, I cohomology be arguments. and could cohomological homology by G the replaced ar- that be group proved could Tate Brauer Hochschi theory 1952, the theory. field how field class showed class of (1952) Nakayama in guments and use have Hochschild should and groups (1950) of cohomology the that sending o l primes all for hr sacnnclisomorphism canonical a is there follows: as this interpret can we Note: a ∈ a aerclsta ri aot14)pitdoti conversations in out pointed 1948) (about Artin that recalls Lane Mac K ∗ otepicplideal principal the to p C L .i h rvos edefine we previous, the in .. of K of /C K K K ◦ and n ucetylremodulus large sufficiently and −→ ∼ C C ϕ  m Gal K nue nisomorphism an induces ( i K ψ a : φ L/K K 7−→ ab K / ( Gal 19 a / /K ∗ C/C Gal ) → . ( ( a ) K ( ) J ◦ L/K  ab τ ≃ 7→ /K C τ ) ←− lim | K L ) , m C I K m m /K n o vr finite every for and , hr sacommuta- a is there ∗ iea elegant an give n some and , bexten- dcase ld groups nding tion his ld n - Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. mn te hns otisannbla ls edtheory field class nonabelian a contains things, other among eeaiaino ri -eisadHceL-series Hecke and L-series Artin of generalization ol epsil oddc hmfo bla ls edter.Weil theory. field class abelian from them deduce to possible be would h orc ttmnswsthe was statements correct the h elgroup Weil The follows.. as stated ht erltr ri adta ehdls at nteeitneo n a of existence the in faith lost had he that theory. said field Artin class abelian later, year a that, oaeincasfil hoya pca ae o oa field local a For case. special a as which theory principle, field functoriality class conjectural nonabelian his a stated Langlands 1967 In n W L Spl rs nti aho.Langlands fashion. this in arise eettosof resentations GL rbnu lmn.TelclAtnmpi bla oa ls edthe field powe class integer local an isomorphism abelian as an in field as map residue regarded Artin the be local on The act element. that Frobenius elements the of sisting pcfiswihautomorphic which specifies yial nnt dimensional. infinite typically nwr h rgnlqeto o l nt aosetnin of extensions Galois finite all for question original the answers G oopi L-series tomorphic ie ycmoiinwith composition by given reuil ersnain of representations irreducible tblzro ahvco soe.O h aossd,let side, Galois the On open. is vector each of stabilizer utpisi yteaslt au of value absolute the by it multiplies nan on N uhthat such qiaec lse fpis( pairs of classes equivalence yJcut ittkiSaio n hlk nteatmrhcsd,an side, side) automorphic Galois the the on on Deligne Shalika and and Langlands Piatetskii-Shapiro, Jacquet, by for under ( ,Lnlns(n unl)hv rvdtecnetr nsm cases. some in conjecture the proved have Tunnell) (and Langlands 2, = K ,ρ s, ab 8 9 saein n all and abelian) is sanloetedmrhs of endomorphism nilpotent a is n S nta fsuyn h set the studying of Instead nteatmrhcsd,let side, automorphic the On • • WEIL LANGLANDS K ( ( b h map the (b) of a h eemnn of determinant the (a) C L/ farepresentation a of ) n set htteei aiyo ietos( bijections of family a is there that asserts φ orsodt eti ersnain of representations certain to correspond ) dmninlcmlxvco space vector complex -dimensional K W Q hnbcmsta fdsrbn h e faayi ucin that functions analytic of set the describing of that becomes then ) otecnrlcaatrof character central the to K 10–98.Dfie h elgop hc nbe i ogiv to him enabled which group, Weil the Defined (1906–1998). agad ojcue httehmmrhssfrom homomorphisms the that conjectures Langlands . GL W σ 13–) h agad rga savs eiso conjectur of series vast a is program Langlands The ). (1936– K 1 n ( K preserves n ojcue hteach that conjectures and , of K = ) ri rvdalArtin all proved Artin , K π 7→ ρ sdfie ob h ugopof subgroup the be to defined is φ K ,N r, L K σ of σ ∗ GL sre rs nti aho.Tu,teconjecture the Thus, fashion. this in arise -series n For . L n Spl only 9 ( Gal r utcaatr,adtecrepnec is correspondence the and characters, just are ( where ) fcosand -factors φ π π a osrce ls of class a constructed has n K ,vee sacaatrof character a as viewed ), V : ) ( S A K ( ( from n uhta conjugating that such L/K rbe:oc eke htte ee it were, they what knew we once problem: L/K > n A π ncmlxvco pcsfrwihthe which for spaces vector complex on ) ( φ 20 K . n K − ( r etesto qiaec lse of classes equivalence of set the be ) K 1 .Tepolmo eciigtesets the describing of problem The ). ,w hudsuyteArtin the study should we ), h ersnain of representations the 1 K sasmsml ersnainof representation semisimple a is ( V σ ) ∗ .TelclLnlnsconjecture Langlands local The ). → rva na pnsbru,and subgroup, open an on trivial , ε notelretaeinquotient abelian largest the onto fcoso ar of pairs of -factors G L n GL σ L sre r uoopi.For automorphic. are -series ( n K S ) ( n n ) ,ρ s, ≥ ( K 1 . satmrhc and automorphic, is ) .For ). G N L n Gal by sre,called -series, ( W K Q K n For . r etestof set the be ) π ( K ( corresponds , K ,terep- the 1, = s(sdefined (as ’s σ GL ( ) ab hscnbe can this common a e /K n n σ W 8 includes ( L fthe of r r can ory (so 1 = ∈ K K sthat, es relates -series con- ) are ) W by d W into au- on- K K ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ojcueisl a rvdb arsadTyo 20) eea otsl months Several proof. (2001). simpler Taylor a and found Harris Henniart(2000) by proved was itself conjecture • o each For • c for (c) (d) σ n χ omtswt asg otecontragredient, the to passage with commutes K ∈ enatsoe hr xssa otoesc aiy The family. such one most at exists there showed Henniart , A 1 ( K ), σ n ( π ⊗ ( χ ◦ det 21 )= )) σ n ( π ) ⊗ σ 1 ( χ ) π 7→ π ∨ ater, Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. h ento fa of definition the ento 3.1.1. Definition h e of set The u lotehge eiaie of derivatives higher the also but ewudlk oitouethe introduce numbers to like p-adic would We of Definition 3.1 valuations of theory The 3 Chapter hssget h ute usin hte o nytevleof value the only not whether question, further the suggests This 3,Mle[]adLn 6.Tevlainter sorsatn on othe the to of point invention starting the our with is started theory which valuation theory The our [6]. of velopment Lang and [4] Milne [3], 91 ihave oitoueit ubrter h oeflmeth powerful the predomin a theory integer theory number the function into to in Hensel(1861- plays associate introduce Kurt which mathematician to expansion, the series view by power century a 20th with the 1941) of beginning the at rpsto 3.1.2. Proposition ersne nteform the in represented Proof. nteseiccs fapstv integer positive a of case specific the In eueidcino n... on induction use we where p ai neesi eoe by denoted is integers -adic a 0 + p a o xdprime fixed a For a ai integer. -adic 1 0 f h eiu classes residue The p ( ≡ ≤ p + := ) f a a i a 0 n ∈ 2 p < + Z o p mod f p = 2 a t vle tprime at ”value” its 1 a + p 0 p ai ubr olwn h da fNeukirch of ideas the following numbers -adic ... f + + a eraoal end hslasu to us leads This defined. reasonably be can p ∈ a ... a , 1 22 κ p + ( Z p p 0 + a p ai nee safra nnt series infinite formal a is integer -adic ) n . ≤ κ , o p mod a ... n − a 1 ∈ + i p ( p, < N n a p − k ehave we , = ) 1 n p p o p mod k ∈ ∈ Z Z 0 = ı /p Z /p ie , Z n n Z , 1 , 2 p ... a euniquely be can ai numbers -adic n oe We role. ant f ∈ Z dof od at de- p , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. eiu classes: residue ssqecsof sequences as Z ota ecndfieadto n utpiainof multiplication and addition define can we that so nlg ihteLuetsre ew tahto attach we ie series Laurent the with analogy ecl ttepai xaso of expansion p-adic the it call we ensa defines r unique. are where n ytepeeigproposition preceding the by and ensasqec frsdeclasses residue of sequence a defines p o if Now h sequence The ntedrc product direct the In vr integer Every ehv h aoia rjcin ewe ieetrings different between projections canonical the have we and noarn and ring a into p ai integer -adic f ∈ a s Q s 0 n f n p − n ewrite we = n oegnrlyeeyrtoa number rational every generally more and Q n m Y f s ∞ =1 1 = p n a Z + ∈ 0 noisfil ffatos ehv oview to have we fractions, of field its into = /p s Z + Z a , 2 /p 1 Z mdp fmod ( 2 a s a p p = .. , 0 n ) 1 − n ←− a , λ p f Z a m = = 1 shsbe end u ahra eune of sequences as rather but defined, been has as 0 + 1 s +1 = = ... , 1 λ s { ν + X Z ∞ .. n =0 n /,g h g, g/h, n = f /p { + / p g/h a ( + o p mod ( ∈ { ∈ s 1 a x a 2 .. n p Z ν a 0 Z n +1 23 + p n ) 0 /p ←− ν n − , λ = ) − a mod ∈ 1 1 mod n n 2 ∈ m m N .,p ..., , p Z ∈ Z n x , Z n , + s − ∈ t ( sth /p Z n p 1 p a p n /p Z 3 2 m − f Z p , ∈ n +1 1 = 1 h -dcnumber p-adic the h p gh, Z ←− Z } λ etc p p /p 3 n ai ubr,wihturn which numbers, -adic , + 2 ∤ 1 = ... n n ehv the have we and 1 = ) ... , .. h Z } } , ∈ 2 ... Q Z p /p n Z f ∈ Z p not Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. mally nteuulmne slmt fCuh eune frtoa ubr.T numbers. rational of sequences Cauchy of by limits proposed as was approach manner usual the in rpsto 3.1.3. Proposition result: our have we and rjcielimit projective eunecnegn o0wt epc to respect with 0 to converging sequence a ihapwro h rm number prime the of power a high epc owihteaoesre ovres htthe that so converge series above the which to respect fara ubr u tde o ovrea h eia rcindoes. fraction decimal the as converge not does it But field number. the Nonetheless real a of ecnsenwta h umnso a of summands the that now see can we . The 3.2 h ersnaino a of representation The orpaeteodnr bouevleb new a by value absolute ordinary the replace to ecnadadmlil opnnws.W hshv the have thus We componentwise. multiply and integers add can we ntesm aho stefil ftera numbers real the of field the as fashion same the in eset we eebe eymc h eia rcinrepresentation fraction decimal the much very resembles h exponent The h rjcielmti urn ftedrc product direct the of subring a is limit projective The Let ecnie ( consider we v p a 0 = (0) = Z ←− lim p . c b ∞ n ,c b, Z p a hsgvstefunction the gives This . x /p ai bouevalue absolute -adic 0 m n fterings the of ∈ ) n + n a Z nti ersnaini eoe by denoted is representation this in Q ∈ hr sabijection a is There 0 Z a N = 1 p + p eanneortoa ubr eetatfrom extract we number, rational nonzero a be ( of ai integer -adic sth { 10 a 1 ( 1 Kurschak J x p + ) p ai ubr a ecntutdfo h field the from constructed be can numbers -adic n λ ) v + Z Z n n p ∈ p /p a ( a x N : 2 −→ 2 ∼ | n n Q ( a p p ∈ +1 Z 10 | 2 1 p → spsil,ie possible, as ←− lim n ie Y = ) 24 ti enda follows as defined is It . + ∞ = ) =1 2 Z p n ... ai series -adic p + Z {∞} ∪ Z 1 m x /p | | /p ... n n n p n Z . Z 0 p 1 = ai bouevalue absolute -adic ≤ 0 λ a a a , ≤ n R i 2 = 0 ( .. , p < nta es ehave we sense that In . x a + p p i n ai ubr appear numbers -adic m v +1 a < hssti aldthe called is set this p 1 Q c b ( = ) 10 p ′ ′ a n ∞ n eptfor- put we and ) + =1 ( igo p-adic of ring b x a ′ n 2 c Z ′ } p p , /p 2 and 1 = ) n + | | Z ... p ,c b, where form with his as Q Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. hc a ecekdt aif h properties the satisfy to checked be can which eas for because -dcaslt value absolute p-adic on rvdsaCuh eunevaisprilsums partial its via sequence Cauchy a provides rpsto 3.2.1. Proposition n a hwta h bouevalues absolute the that show can One where example: satisfying bouevalue absolute o hsw define: we this For frtoa ubr uhta o every for that such numbers rational of Q where nve fte3poete bv tstse h odtoso a of conditions the satisfies it above properties 3 the of view In • • h function The • A ewn o ogv natraiedfiiinfrtefil of field the for definition alternative an give to now want We n ute omi power a is norm further any , v v v acysequence Cauchy p p p p ( ( ( aisoe l rm ubr swl stesymbol the as well as numbers prime all over varies a ab a = ) x + = ) vr omlseries formal Every | + m > n x b ) n ⇔ ∞ ∞ | | v ≥ − p ( = min v x a sdntdby denoted is p + ) m n has one a ∞ | scle the called is 0 = p | | o vr ainlnumber rational every For { | v v x = , p p p n ( ( sgvnby given is | ∞ : b a − ihrsetto respect with ν ) n X Q ) = ν X − , x + ∞ =0 m v 1 → m p ∞ a ( | | | a p b ν R x ν ) p = o l ,m n, all for ε < n p } Y ∞ ν -dcepnnilvlainof valuation exponential p-adic p ν a , | = | | ∞ hsgvstefollowing the gives this , p | ≤ a n ν X 25 p s | | and | =0 − p > ε max or 0 1 →| 7→ p 1 = a ≤ | | and ν | | ∞ ,teeeit oiieinteger positive a exists there 0, p a a m p ν ν | s ≤ p x > sb ento sequence a definition by is | | p < o oereal some for ν s . p p ai numbers. -adic 1 m .Teusual The 0. norm Q The . on { x Q n n } 0 . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. localization l nt subextensions finite all ftevaluation the of of tetnsi nqewyt continuous a to way unique a in extends It hr ntecs fa nnt extension infinite an of case the in where of restriction by obtain we values eembed we h mapping The sasqec ovrigt nteuulsense. usual the in 0 to converging sequence a is n h nqeetnino hslte auto to valuation latter this of extension unique the and ideal to value h osatsqec ( sequence constant the closure edn ftecoc ftesequence the of choice the of pendent o vr valuation valuations every For of Extensions 3.3 p ai nullsequence -adic L Q sequence A Let hslmteit because exists limit This ring a form sequences Cauchy The p ihrsetto respect with m | | ygvn h element the giving by | | n edfietefil of field the define we and , L/K K v v v of Q resp of K ea leri xeso.Cosn a Choosing extension. algebraic an be in of τ K v { | | : x hnw banon obtain we then , Q L x v v L h aoia xeso of extension canonical The . n = p v ihrsetto respect with to } → on { yascaigt vr element every to associating by w v in y w L K n − K u h union the but , of ,a ... a, a, nohrwrs if words, other In . ∈ } Q L v lim , i /K K sovosycniuu ihrsett hsvaluation. this to respect with continuous obviously is scle a called is K | v m x v n x | ecnie h completion the consider we to {| →∞ p .The ). resp of satisfies = x := τL n L/K τ { x | | τ Q x p lim x w K n p w : } | nextension an n : ai ubr ob h eiu ls field class residue the be to numbers -adic p w hn[ When . L L 7→ v nullsequence p } L = saCuh eunein sequence Cauchy a is hsuinwl ehneot aldthe called henceforth be will union This . := where , w ai bouevalue absolute -adic n { = 26 lim mod h utpiainvaluation multiplication the →∞ → L x x τ v → L/K R R n | w τx ◦ } n K h uleune omamaximal a form nullsequences the , / | K τ = →∞ := m x m K ihniscasmod class its within | v v v n , v L | | S -embedding | ∈ L v resp p | : y i w − v v R ∈ K n L osntma h completion the mean not does xed rcsl h absolute the precisely extends / | to lim iw R p ] m v < ihrsetto respect with K K 0. = a K ftecompletions the of r ie yteabsolute the by given are h bouevalue absolute the n -embedding v ∞ ∈ →∞ v by , saandntdby denoted again is Q τ K | | x τ v h eiu ls of class residue the v sgvnb h rule the by given is . p n n nalgebraic an and R on m n ti inde- is it and Q eas any because | | sextended is p if L | iw x n of | v p Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. inand sion h olwn eutgvsu opeedsrpino h possibe the of description complete a v us gives result following The hoe 3.3.1 Theorem h bv eddarmi fcnrliprac nagbacnme theory number extension algebraic global in the from importance central passage of the shows is diagram field above The h composite the nqeetnino h valuation the of extension unique new a field hnteaslt auscrepnigto corresponding values absolute the then opeeextension complete Proof. n hsrpeet n ftems motn ehd fagbacnumbe algebraic of methods important most the of the one theory, represents thus and eas if because over over conjugate where has o some for of in sequence by to v σ ( h aoia xeso ftevaluation the of extension canonical The ( eswta every that saw We ( o vr automorphism every For . v i ii ii L K ◦ ) L = Let ) ) . K { τ vr extension Every v esalpoetescn statement.. second the prove shall we n hrfr a ob t opein If completion. its be to has therefore and x w extensions Two Since . v -embedding r hrfr h same. the therefore are n v K ◦ } L/K σ auto of valuation a τ n oa-ogoa principle local-to-global -embedding K ∈ and n thus and N v K ent eeta h sequence the that here note We . v sa is sfiiete h field the then finite is v ( σ steol xeso ftevaluation the of extension only the is xeso Theorem Extension ◦ L τ w τ τ ′ Cuh eunein sequence -Cauchy K with v = τ w · ◦ v -embedding K : σ τ K ◦ ftevaluation the of L v ◦ σ = | τ Then: . x τ of → ∈ | v and w of σ K L Gal K ◦ = ∈ L L K v ( v L −→ v σ ecnie h iga ffields of diagram the consider We . τ twl esi ob ojgt to conjugate be to said be will It . v Gal q w ( n ◦ from K ◦ τ K τ = | N 27 τ v .Teetnin nue to induced extensions The ). : . ( v /K K L K ′ K L L L ) −→ K w σ r qa fadol if only and if equal are . v → L v K v /K /K v L v Let etoebdig of embeddings two be ) w v v K L/K and v K rssa h composite the as arises oteextension the to v n hence and v w ⊆ v ( of ) x v L/K ) from L w aeu nextension an us gave | otelclextension local the to v w x τ K L aif h relation the satisfy i opee otisthe contains complete) (is ea leri edexten- field algebraic an be w v L n /K over { v ovre ntefinite the in converges to x τ v from a degree has L K n w } L v n w speieythe precisely is ∈ eoti with obtain we K /K N v τ xesosof extensions L a to L v and τ ehave We , w conjugate v w by over -Cauchy K < n = L = τ v w τ one , ′ v v /K and K It . are ◦ ∞ ◦ v τ τ v r . , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. K fasequence a of o oesequence some for in Indeed, limit a othat so a n let and h sequence the K v τ Conversely: xd Extending fixed. v ′ L -isomorphism · τ σ K τL v and : lal h correspondence the Clearly . τL sdnein dense is τ { τ → ′ x x ′ Let n n r nedcnuaeover conjugate indeed are τ } x = ′ n n ilsa isomorphism an yields and L ,τ τ, σ στx hc eog oafiiesbxeso of subextension finite a to belongs which ethe be τL oa to ′ n : · ovre oa element an to converges K L K σ K v v → σx : oeeyelement every so , -automorphism -isomorphism x τL K = = v · lim lim K etoebdig uhthat such embeddings two be v 28 n n → →∞ →∞ x 7→ τ K ′ τx στx L σ σx v σ . n · τL = K n ∈ osntdpn ntechoice the on depend not does x τ v · Gal ′ K ∈ ◦ v τL τ ( −→ − K σ 1 · v ecnextend can We . K τ /K ′ v L L v a ewitnas written be can · As . gives ) K v v hc leaves which v ◦ ◦ τ τ τ ′ = = = v v σ σ ◦ ◦ ◦ τ to τ ′ τ ′ , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. e hoywihwssgetdb h hoyo uvsoe fin over curves of theory the by suggested was which theory ber tldt h ento fanwagbacsrcue ” structure, algebraic new Abstr a of fields. function definition and the fields to th number both of led theory to it field common class is global law and reciprocity cal the formation behind algebra class cohomological The of Definition 4.1 theory field class Abstract 4 Chapter h oa n lbldaiytere,adtechmlg fagbactr,but example: tori, For algebraic ou came. of worked directions. soon cohomology be new They the to for and time things theories, t was few duality field it a global class still and deve abelian were local two one-dimensional the There classical those the maturity. With the that full embodying say existence. conceived reached to transferred Weil formation their fair time, proved be class same is and of to the it groups notion about 1930’s Weil At the the of theories. to idea of both led theory of and features field enabled theory, common class It global light. local the new cohomol the to shed the Tate of of and theorems use Artin Nakayama, systematic many corresponding Hochschild, impl the the by which 1950 groups isomorphism and Around of an group laws. by reciprocity Galois isomorphic, known canonically abelian all and are corresponden an field, group one that that class to of finite later ideal groups one the years class natural that ideal several in generalized proof proof develop the are Takagi’s of of field with quotients century number the 1920’s with a a in of roughly came extensions of point i abelian result high and the The groups) is profinite 1850-1950. cover of we definition mathematics rigorous oth- a The among [5]. (for found Tate [2] be and and can Artin [1] information locally, relevant th Neukirch The groups in in multiplicative ers formation. is the class difference and a The of globally axioms classes theories. idele our the of that features common the embodies 1 epl’ conjecture Leopold’s program Langlands theory the Iwasawa and laws reciprocity theory Non-abelian field class dimensional Higher IWASAWA 11–98.H nrdcda motn e prahit a into approach new important an introduced He (1917–1998). 1 29 ls formation class t fields ite gbacnum- lgebraic aif the satisfy er had heory ,sc as such t, lopments ,which ”, proofs e Artin’s acting ment, lo- e ogy the ied ce n Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. a h rpryta ftesubgroup the if that property the has nomto bu h oladfnto ftetheory. elegant the and of comprehensible function easily and an goal in the sp gives about the and information from theory, theory, field field of class erations of characteristic is which machinery, h btatvrino h antermo ls edter.Tento of notion The itsel theory. Theorem field Tate’s class of view theorem can main formation one the class of fact version in consi abstract essentially Theorem; the which Tate’s above in mentioned sumptions theories concrete the for bv h prime the above clioopimbtenteGli group Galois the between isomorphism ical htw hl s ae n ae norter fvaluations. of theory our on based on, later use shall we that hss aldrcpoiylwi h antermo ls edthe field class of theorem main the is law reciprocity called so This iea lgn eomlto fcasfil hoy ihLubi With fields theory. local field of class extensions of abelian reformulation generating elegant an give extension rnil ncmo.Alo hs hoisivl aoia bijectiv canonical a field th invole a have theories of justified, extensions these abelian similarly of the is All between theory spondence common. field in class name principle the which for modules orsodn module corresponding a by L h niiulpie stefnaetlpicpebhn class behind principle fundamental the is primes individual the h rniinfo h lblextension global the from transition the to 2 F is ewl aeasalprnhsst nrdc h decomposit the introduce to parenthesis small a make will we First hsmi hoe a etakdbc oacmo ytmo axioms of system common a to back tracked be can theorem main This bla adnon-abelian) (and Abelian If e’ osdrtepiedecomposition prime the consider Let’s oa n lblcasfil hoy swl sasre ffrhrtheories further of series a as well as theory, field class global and Local theorems). (now conjectures Serre conjectures Stark Lubin-Tate eilsrt hsdarmaial sfollows.. as diagrammatically this illustrate We TATE K contains L/K ilstevlaino h field the of valuation the yields L/K 12–) epoe e eut ngopchmlg,wihal which cohomology, group in results new proved He ). (1925– safiieetnino ubrfield number a of extension finite a is K 2 te”ls edascae with associated field ”class (the p oa hoy ae xlctter o ucinfils Drinfeld fields, function for theory explicit Hayes theory, local p ic h etito ftevlainascae with associated valuation the of restriction the since , sbsdo hsie.I eaae h ueygoptheoretic group purely the separates It idea. this on based is of K hnw write we then , A K soitdwt h field the with associated l ai representations -adic L K p = F 30 F I e K .. | L/K p ⊆ F nti aetecompletion the case this In . L K soitdwith associated ′ A e F G ′ p K L/K otelclextensions local the to orsod oa bla field abelian an to corresponds I ) hnteeeit canon- a exists there then ”), K n h atrgroup factor the and K efuda xlctwyof way explicit an found he n and n eti ugop of subgroups certain and K hscorrespondence This . F p . rm of prime a edtheory. field t fteas- the of sts cfi consid- ecific ory. oe i to him lowed following e o group ion L F corre- e L L F | A F K lying K from of p way as f /I at L a . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. from of σa ne hsisomorphism, this Under npriua if particular In n hrfr neeeto h aosgroup Galois the of element an therefore and if fact In (resp bv uoopimi h otnosetnino h automorphism the of extension continuous the is automorphism above Let ehv that have We σ otecompletion the to ovreyeeyatmrhs of automorphism every Conversely etito to restriction sa isomorphism. an is n ecnietf hs w rusi h olwn text. following the in groups two these identify can we and hoe 4.1.1 Theorem as: stated is which [1], rpris o ahsubgroup each For properties: ∈ i p hr sacanonical a is There If u anrfrnetwrstepofo eirct a sTt’ t Tate’s is law reciprocity of proof the towards reference main Our • • G ∈ L/K in H H L a hnif then , L F L 2 1 eeae h group the generates ( ( .If ). then , ovre in converges ,A g, ,A g, a efiienra edetninwith extension field normal finite be ∈ ) 0 = ) F L L K F a F scci forder of cyclic is ow aeacnnclisomorphism canonical a have we so , σ ustruhaltepie of primes the all through runs p ˆ p | ie , = p F = scnandin contained is ( L ⇒ = F aesTheorem Tate’s F F ˆ a − , e F σ = L where , G hnw aethe have we then , F i a lim σ | K F F a F p K − p ∪ stedcmoiingopof group decomposition the is n esay we and wrt -isomorphism p H X lima : i F sfie elementwise. fixed is | H ∈ 2 g p p ˆ ( σ [ q ,A G, ⊆ L L or ( G | F L ( i g F G, F n h aoia smrhs sobtained is isomorphism canonical the and , L F | G F o oesequence some for L 7→ F : G = ∼ F swl as well as ) Z F ˆ K hntemap the then , ) −→ L −→ ) σ eoe h rm da ftefield the of ideal prime the denotes ) 31 . σ σa G σ F p → F [ = ] /K L suethat Assume L L F = K scnuaeto conjugate is σ p H /K F F p σ L ilsa uoopimin automorphism an yields q -automorphism G p +2 F L : L L − K F ( G σ over ,A G, /K F i σa lim ] = since p ie , a ) Gal p A i hnw have we then , ∈ i p F is ( F ∈ L L/K isunder lies over G wrt hntesequence the then , G L F σ mdl ihthe with -module F (= ) ⊆ σ K . G . L F F G /K and L/K heorem p σ G The . F of ,if ), σ K by F L p . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. h opstmo uhfields such of compositum the G ugopwt h index the with subgroup of subgroups open ovrey lsdsbru ffiieidxi pn eas ti th is closed. it is because complement open, its is hence index cosets, finite many of finitely subgroup closed a Conversely, at ntl ayo hs oescvrtegroup the cover cosets these of many finitely pact, group G fteyassnete15s h on fve fcasfil hoyhsslight has theory field class of of groups view Galois of the point describes approach the classical 1950s, The the changed. since years the of nti eto ontueti nepeain h pnsbrusof subgroups open The notions interpretation. abstract this the use although not extension, do field section Galois this infinite in an of topology) eoeivsiae h e falfiieetnin ftefield the of in extensions groups finite Galois all infinite of consider to set is the essenti the view investigates An of field. one point ground ie modern global the or of local the ture of invariants arithmetic using formally oooy emytikof think may We topology. refere without modules, their and concrete background. groups with be theoretical profinite always can number of but results terms itself develope many in Nevertheless, in been solely end mind. has an in groups” applications as profinite theoretical not of thei theorists, ”algebra in number interest extensive of by ignor an objects to reason as groups, this useful top profinite For investigate the be to type, and to and this motivation of compact proves groups theoretical Hausdorff, It number are their moment, they groups. the which topological under disconnected topology, locally Krull the topology, a pnsbru steuino oe)cst,tu pn n since and open, thus complem cosets, the (open) fact, of In union index. the finite is of subgroup subgroups open closed the precisely K L/K h extension The nextension An ie rfiiegroup profinite a Given h ”field” The esalnwitoueasrcue ethe ie structure, a introduce now shall on.. We later theorem this of use make shall We Let If . absolute = L/K G G K eapont ru,i opc ru ihtenormal-subgroup the with group compact a ie group, profinite a be K ⊆ /G snra,te h aosgopof group Galois the then normal, is aosgroup Galois L K L n en h ereo uha extension an such of degree the define and 0 L/K scci,aein ovbeec edfieteitreto and intersection the define We etc. solvable abelian, cyclic, is L/K G with etecoe ugop ffiieidx elblec such each label We index. finite of subgroups closed the ie scle ylc bla,slal t,i t aosgroup Galois its if etc, solvable abelian, cyclic, called is G scle omlif normal called is K K G 0 n alteeidcs”fields”. indices these call and , K G = [ G L hs rusitisclycm qipdwith equipped come intrinsically groups These . ecnie h family the consider we , K G G : steGli ru edwdwt h Krull the with (endowed group Galois the as i L/K K K scle h aefil.If field. base the called is ysetting by ( = ] = = 32 i \ =1 G n G K K K G /G : i L G L rfiiegroup profinite L ⊆ L/K ) G G K ec h ne sfinite. is index the hence , sdfie stequotient the as defined is sanra ugopof subgroup normal a is { G K G L/K L : finite ⊆ K ntecourse the In . k as G toc,via once, at ∈ w right. own r K formulated extensions c othe to nce G no of union e ewrite we , X n fan of ent number ological } scom- is lfea- al G stead, ,for e, fall of are ly d Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. omlextensions normal ( pair the the K if oueascae with associated module rtda iceemodule) discrete a as preted ento 4.1.2. Definition rvoseuvln odtos h pair the conditions, equivalent previous nti otx ti motn oke h oooia tutr on structure topological the keep to important is it context this In G h cinof action The tsol aif n ftefloigeuvln conditions: equivalent following the of one satisfy should it h utpiaiegroup multiplicative the utpiaieywitn Let written. multiplicatively n a s ta noinainfrwa follows. fiel what class for local orientation in an play as into it comes use that may example one this precisely is It if oin,w banfrec rfiiegroup profinite each for obtain we notions, sions nee ytesto fields of set the by indexed , G ∗ If . If ecnie o o ahnra extension normal each for now consider We ntecasfil hoyeapemnindaoe eovosyhave obviously we above, mentioned example theory field class the In • If • • e ( Let ntefloigw osdrmodules consider we following the In If K G G i)Freach For (ii) (iii) i h map The (i) L/K G G L/K L/K s(oooial)gnrtdb the by generated (topologically) is K K L steGli ru fa(nnt)Gli extension Galois (infinite) a of group Galois the is ,A G, ′ = -module ,A G, A ⊆ = and sanra xeso,then extension, normal a is T = L σG eafrain ntefloigw hn ftemodule the of think we following the In formation. a be ) i n then , omto,w aial enb hstefraino these of formation the this by mean basically we formation, a ) =1 G S A L L U K on G ′ σ /K A a G A − L/K K = If U 1 A L ∈ A i × o ipiiyo oain eset we notation, of simplicity For . A G for ojgt ncase in conjugate K where A hudb nacransnecniuu.Mr precisely, More continuous. sense certain a in be should A G K oehrwt the with together sapont ru and group profinite a is ⊆ N K h stabilizer the σ → H ∗ A ie = ∈ q ftefield the of A L ( U { { G L/K . G a ih( with X hnw write we then , ustruhalteoe ugop of subgroups open the all through runs K ∈ o ahfield each For . K : = ) A K = : ,a σ, σa 33 ∈ H { i Y =1 ( N σ n L A q X ,A G, ) ′ G = A ( ∈ n h ar( pair the and , L K 7→ G G } = K G o l σ all for a G nwihapont group profinite a which on L/K i L/K etefml foe ugop of subgroups open of family the be sa is i ) σL σa omlGli theory. Galois formal a : in scle a called is L L/K σa A , -modules A ′ scniuu (here continuous is G o some for G K = L/K = and , L sa is ) ∈ σL h ooooygop of groups cohomology the a } X mdl.We ecall we When -module. G n ecl w exten- two call we and soe in open is ∈ ,N G, mdl aifigthe satisfying -module formation ecnie h fixed the consider we A σ N/K G L ∈ K . ∗ } G saformation. a is ) then K hoy and theory, d ihthese With . G G A . G nmind. in sinter- is G G cson acts A A acts. K as = Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. phism module isomorphism smrhssaecmail n eoti neuvlnebtenthe between equivalence an obtain we and compatible are isomorphisms G intefis ooooygopvanishes: group cohomology first the sion ilstehomomorphism the yields nioopimbetween isomorphism an ensa smrhs between isomorphism an defines nafil omto ehv that have we formation field a In hti,freeyinteger every for is, that oml hntesequence the then normal, N ecl omto ( formation a call We sn h qiaec ftemodules the of equivalence the using If If nohrwords other in nadto eas aetersrcinadcrsrcinmaps corestriction and restriction the have also we addition In eew nyne oasm that assume to need only we here seatfor exact is ⊆ L/K K G A L ⊆ L H ⊆ snra and normal is H n the and L q ( σ q G N/K ( ⊆ ∗ q K G 1 1 omtswt nain etito n corestriction. and restriction inflation, with commutes n xc for exact and 1 = L/K N with → → H ) H G satwro omletnin of extensions normal of tower a is q H −−−→ H A , res ( H q σL/σK G ( 2 q G L L q L/K ( ( N/K ,A G, ( L/K L/K q N = ) A G σ H H homomorphisms L N/L and ∈ q -module A , q ) ( H and a ) ( ) ) L/K G τG −−−→ N/L inf q −−−→ −−−→ N A , inf inf G ( H then edformation field G N ) G L A L N N N −−→ 1 L/K ) L/K ) res 7→ > q σL ( a ) H omlin normal −→ L/K σ H H A −−→ n H and 7→ ic ( Since . cor ∗ q στσ 34 H A , σL ( 2 q N/K and if 1 N/K ( ( H σa q N/K N/K A 1 = ) hsevery Thus . N G H ( − q L G N/L H 1 ( G q σL/σK G q N/L ( and ) snra.I both If normal. is i ( σL/σK G ) ) στσ ( σL ) N/L G N/L N/K −−−→ −−−→ −−→ res res A , inf K o q for A − hnfrec omlexten- normal each for when L L ) N σL n h ooooytheory cohomology the and 1 for 1 = ) A , ) H and −−−→ G ) cor H H q ti ayt e htthe that see to easy is it N σL σ q 2 ( ≥ K G ) ( ( and K ) ∈ N/L N/L 1 σa N/K H G ehv inclusions have we q i = ( ) ) ilsa isomor- an yields A , 1 = N/K σ N ( N τG . q .., , ) ) and L ) − a G these , L 1. L/K are - Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. H formation. and sion l fw aeti dnicto n tpfrhr If further. step espe one formally identification become this will take ideas we our if of ple presentation The injective. is sawy exact. always is xesosof extensions npriua,if particular, In epc oteiflto as etu aetedrc limit direct the take thus We maps. inflation the to respect i h netv nainmp.I eietf hs ruswt t with groups these identify we then If embedding, this maps. under inflation injective the via eepaieta h nainmp r ob nepee sinclusion 4.1.3. as Remark interpreted be to are maps inflation the that emphasize we aiga ohisthe cochains as taking sfrfiiegop,w a en ooooygroups cohomology define can we groups, finite for as n eoti group a obtain we and ie n extension any Given nfc if fact In ensa element an defines group the in contained 2 ( If esalseso htwhen that soon see shall We L/K K ⊆ sebde ntegroup the in embedded as ) A c L h utpiaiegopo h xeso ed hnw aeafield a have we then field, extension the of group multiplicative the ∈ ⊆ K H N hntegroups the then , K 2 Let ( r omletnin,te ecnawy hn ftegroup the of think always can we then extensions, normal are ⊆ H /K G q K L H ( H K continuous res G ) H ′ ⊆ 2 hnteei nextension an is there then , /K 2 K ( eapont ru n let and group profinite a be 2 ( H L/K H N K H ( H A , H L/K /K ′ 2 2 of 2 H 2 c satwro omletnin of extensions normal of tower a is ( 2 ( ( ) ( L/K ( ∈ L/K 2 ) = ∼ K nwihaltegroups the all which in ) /K /K ( ) ⊆ /K H L/K ec h etito map restriction the hence , eoti aoia homomorphism canonical a obtain we , H G H 2 H ) ) lim = ) ) q 2 ( = ) steGli ru faGli edexten- field Galois a of group Galois the is maps −−−→ −−−→ ( ( L/K 2 res res −−−→ H eoesbrusof subgroups become ) L/K inf ( 35 N/K /K −→ K K 2 [ N L ( ′ ′ ′ N/K L lim = ) ) G omadrc ytmo ruswith groups of system direct a form ) H H H H H ⊆ ) K 2 2 2 2 ⊆ 2 ( ( −→ H ( ( × ( ,sneteiflto map inflation the since ), L/K N/K L/K L/K H 2 /K ... L ( 2 H × ( ) /K ′ ′ ) A ) ) ) q K G /K L ( ea be L/K ′ H K H ) ⊆ agsoe h normal the over ranges ) q 2 → ( ( K G G L/K ) A ′ K K H Then . ⊆ A , mdl.Exactly -module. 2 K ( r embedded are ) L ) ehave we , /K othat so , for erimages heir ilysim- cially ,and ), q here. s ≥ 0 c by is Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. aifistefloigaxioms: following the satisfies o vr omlextension normal every for ento 4.1.4. Definition e we map. which invariant subfields, the and on conditions fields compatibility extension certain to imposing passing when ”correct” N xsec facnnclioopim h ocle rcpoiymap” ”reciprocity called so the isomorphism, canonical a of existence a oc h xsec fsc nioopimi btat yimposin by , abstracto in ( formation isomorphism our an on such conditions of following existence the force can the nain map invariant smrhs between isomorphism isomorphism cnncl eirct a,w replace we law, reciprocity ”canonical” ntecoc ftegnrtrof sinc generator isomorphism, the this of to choice arbitrariness the certain on a is there However image L/K • • hs osdrtosla st the: to us lead considerations These h udmna seto nbt oa n lblcasfil hoyi the is theory field class global and local both in assertion fundamental The • fti od,te h u rdc ihagnrtrof generator a with product cup the then holds, this If • nain map invariant xo I Axiom xo II Axiom b If (b) a If (a) A [ L L = : 1 K K K N ] G ⊆ ⊆ + L/K H hc nqeydtrie h element the determines uniquely which , L o vr omlextension normal every For L Z 1 ( h rca on eei htti element this that is here point crucial The . A ⊆ ⊆ L/K .H I. I H II. formation A L ihtefloigproperties: following the with N N H inv stenr ru of group norm the is inv 1 = ) satwro xesoswith extensions of tower a is 2 satwro omletnin,then extensions, normal of tower a is 1 ( L/K ( 2 L/K N/L inv L/K L/K ( L/K G G L/K o vr omlextension normal every for L/K ab L/K ab : ◦ n h ylcgroup cyclic the and ) H 1 = ) res where ) 2 ( = scci forder of cyclic is H ,A G, ( = ∼ = ∼ L L/K inv 2 A A [ = ( 36 L/K K K ) G ,A G, N/K ) L /N /N II. L/K ab sb alda called be is → : .Teeoeadi re ogta get to order in and Therefore ). A L/K L/K :If ): K | H ytecniinta hr san is there that condition the by L/K [ L steaeinzto of abelianization the is L ] 2 eas fTt’ hoe,we Theorem, Tate’s of Because . ( · A A L/K : 1 inv L/K L L K hr xssa isomorphism: an exists there ] N/K ) Z [ L N/K / [ sayetnin then extension, any is L Z : ls formation class u K : 1 L/K L/K K ] oml then normal, ] H Z ∈ 2 / fil formation) (field ( Z H L/K u h so-called the , 2 L/K tdepends it e ( L/K G ie an gives ) sr by nsure L/K remains with ) the g fit if and Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. element ento 4.1.6. Definition hnteei nijciehomomorphism injective an is there then ehv diinlfrua o h oetito map corestriction the for formulas additional have we map rpsto 4.1.5. Proposition diagram scle the called is udmna lse fdffrn edetnin r eae,ie 4.1.7. related, Proposition are extensions field different of classes fundamental o ecndsigiha”aoia”gnrtri ahgroup each in generator ”canonical” a distinguish can we Now • • h xeso rpry( property extension The nodrt aetepoet b biu,w iulz ihtecommu the with visualize we obvious, (b) property the make to order In • • • • rmtebhvoro h nain a ecie n4 in described map invariant the of behaviour the From σ (a) (b) (b) (c) (d) (a) ∗ . u inv res cor inv σ u L/K L/K ∗ ( K L σN/σK N/K udmna class fundamental u ( N/K ( ∈ u u ( = N/K N/L H ( cor = ) u 2 ( ( σ N/K Let ( = ) = ) L/K K ∗ Let Let u c c = ) σN/σK = ) L/K ) u [ u inv H H N ) N/L K K N/K H : 2 uhthat such inv inv 2 inv L ( ⊆ ⊆ L/K ( Ia II ] 2 N/K N/L eanra xeso.Teuiul determined uniquely The extension. normal a be (  ) N/K K L L res N/L [ L ( /K fteivratmpipista if that implies map invariant the of ) ⊆ ⊆ : : u L K ) ) of H L/K c c ] N N = ) if inv inv for 2 L/K ( L/K eetnin with extensions be eetnin with extensions be 37 = ) N/K N/L [ /K σ L / / . ∈ for for [ H [ [ N ) N L snormal is G → 2 1 1 : 1 c c ( : : L/K K K L ∈ ∈ Q  ] · ] ] / [ H H Z L Z + Z : / ) 2 / 2 K Z ( ( Z Z ] N/L N/K cor N/K N/K ) ) . 1 and n h conjugate the and . ,w e o the how see we 5, oml then normal, oml then normal, σ H ∈ 2 G ( L/K tative ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. map ie ytecppoutwt h udmna class fundamental the with product cup the by given ewe h blaiaino h aosgopadtenr residu norm the and group Galois the module. the of abelianization the between nepeain eas ftecnnclisomorphisms canonical the of because interpretation, hoe 4.1.8 Theorem formations ilsacnnclisomorphism canonical a yields hoe 4.1.10. Theorem following the obtain we so and and a oimdaecnrt plcto,hwvrfor however application, concrete immediate no has case smrhs naldimensions all in isomorphism oolr 4.1.9. Corollary h isomorphism The o ecnapyTt’ hoe,t bantemi hoe fclass of theorem main the obtain to Theorem, Tate’s apply can we Now For ic ed o aeacnrt nepeaino h groups the of interpretation concrete a have not do we Since q 3 = q 1 = , ,o eeal o l ooooygop fhge iesos 4 dimensions, higher of groups cohomology all for generally or 4, , eimdaeyobtain immediately we 2 ( anTheorem Main H Let 3 u u ( θ L/K L/K L/K L/K L/K θ L/K ∪ H ∪ 1 = ) nteTermi called is Theorem the in G : 0 : eanra xeso.Te h u rdc map product cup the Then extension. normal a be H ( L/K ab H : L/K eea eirct law: reciprocity general q q G − ( . L/K ab 2 G n H and = ∼ ( = ) L/K G ) . H L/K → Let 38 − A , 2 Z K A 4 , ( ) L/K G ( Z K /N L/K → ) L/K /N → L/K H L/K eanra xeso,te the then extension, normal a be ) , H q Z +2 = ∼ A 0 ) L ( A ( χ L/K aaaamap Nakayama L/K L ( q G u = L/K L/K ) ) − ) ehv uhan such have we 2 ∈ H 2 ( H L/K q ru of group e ( . L/K ) san is in ) . 1 . 9 Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. rmsand primes e iha da in ideal an with fied omls hc stebsso u hoy o h da fti hpe w chapter this of 5.1.1. definitions: ideas Definition following the the For with begin theory. can We our [4]. coh of recipr intuitive Milne basis and Artin mainly beautiful the follow stating our is to ideles, return which of after formalism instead theorem shortly main can ideals, our and how of case see that terms to map parenthesis in small stated reciprocity a make be or can we Artin point this At of Definition 5.1 of ideals terms in Theorems Main 5 Chapter sth: modulus A n writes: one modulus A m • • • ( )if b) )if c) a) p ) m ≤ ( p p p n m ) ( scmlx then complex, is sra,then real, is p 0 ≥ ) m sapouto oiiepwr fpieiel,i a eidenti- be can ie ideals, prime of powers positive of product a is m 0 ∀ p = o l primes all for a ewritten be can . A Q O modulus p K p . m m ( m ( p p ) 0 = ) : m ssi odvd modulus a divide to said is { ( rmso K of primes p for p m 0 = ) m and = or K = Y 39 p safunction a is m 1 m ( ∞ p p m 0 = ) m ( p 0 ) → } where , o lotall almost for Z m ∞ n = sapouto real of product a is Q p p p n ct a in law ocity ( p ) if omology can s e Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. m dasof ideals fpie in primes of with rm ideal prime σ rm ideal prime a hrfr o any for therefore o n nt set finite any For aetehomomorphism: the have the with ent that note we . ( = h quotient The Let fw set we if .hr edefine we ..where o any For where Let o modulus a For lblAtnmp(rrcpoiymap) reciprocity (or map Artin global p L/K , L/K S K ( m π ,gnrtdb rm dasntin not ideals prime by generated ), = ) a F sapieeeeti h completion the in element prime a is K of ) eafiieaeinetninwt aosgroup Galois with extension abelian finite a be p ∈ of of { K L rmsdividing primes C L a m, K S m m yn over lying ∈ ord nqeydtrie ytefloigcniin o every for condition: following the by determined uniquely 1 ( define , fpie in primes of hti naie in unramified is that := a K n rm ideal prime and J a ord 7→ p p m, S ord J ( > a S p 1 1 ob h ugopof subgroup the be to ( ( p h da ( ideal the , − p m 0 a ψ 1 n in ( K ) 1 a L/K − 1) /i p .. = ) m , i p ( 1) ( ≥ , σ K 1 n 1 O : K s : F K ≥ ord m ob h e of set the be to m p J 7→ / m , = a m ( 1 S otiigalpie htrmf in ramify that primes all containing p m p } scle the called is ) , 7→ p m 1 ) Y F ( p → a 40 (( ( p i → | ⇔ ) p m and ) a ( ) ( ∈ Gal a ) L p − J ∗ π ) ord i hr saFoeisautomorphism Frobenius a is there , J l finite all l real all L/K , S m = ∼ )+1 0 = 1) + 1) σa S ( ( ( m L/K p ( p S ( m ) ( ) O | ≡ a ˆ ) . where , ( . J p a ) − K / a a n p (= ) ry ls ru modulo group class (ray) p ˆ N i p 1) − ∈ m p p of ( | K J 1) mod > p m p K ) K S | ∗ ) m = 0 ⇔ gopo fractional of (group ) ∗ sth , sayfie nt set finite fixed any is at F G . ord p ealta,for that, Recall . . p 1 n so and (1) L we , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ψ disamodulus a admits oolr 5.1.3. Corollary 5.1.2. Proposition h group The so L Proof. Proof. andi h enlof kernel the in tained Nm in aiyin ramify o some for o every for S naie over unramified hoe 5.1.4 Theorem ψ isomorphism nta case, that in hntedarmcmue ( commutes diagram the then , ( etesto rmsof primes of set the be i : ψ S ( hsteAtnmpidcsahomomorphism a induces map Artin the Thus Let K K L/K J where , ′ S m /K Let ecntk ntepeiu diagram previous the in take can we , S 1 → a o eijcie od o egta smrhs sfollows: as isomorphism an get we so, do to injective, be not can )=0 Thus 0. = )) ( m F eafiiesto rmsin primes of set finite a be p L J ′ p Gal with = ) yn over lying K S ′ S n lotesto rmsof primes of set the also and m /Nm { ∈ ( sayfiiesto rm dasof ideals prime of set finite any is sa is p L/K p S f ( ( rm daso K of ideals prime si sacpe obe. to accepted is it as J ( eirct law Reciprocity p o vr nt bla extension abelian finite every For ( m K ′ J S enn modulus defining fteeeit modulus a exists there if , / ) ) ( L Let S p m ) ψ ψ ⊇ p sifiie(eas nntl aypie ontsplit), not do primes many infinitely (because infinite is ) disamodulus a admits ) u hsi rpryo h rbnu lmn for element Frobenius the of property a is this but , L/K n ehv oso that show to have we and /i ψ disamdlsi n nyi tfcosthrough factors it if only and if modulus a admits S L K ( L/K ( F hslasto leads This . J K ea bla xeso of extension abelian an be ψ J L/K , K S aiyn in ramifying : K m L/K S  ′ Nm ( J , p 1 K S f ) S ( /Nm · : p ) ψ ψ ψ Nm f ′ sdfie below) defined is J L/K / L/K ( : p p S ) K J ) ′ / . / / = ) ( ′ Gal ′ → 41 ( Gal S J p } esalsyta h homomorphism the that say shall We . J Let ) yn vrapieideal prime a over lying , L S for → L S Gal ( = L ) ψ m ( ( ( L/K m hnteAtnmap Artin the Then . L/K → L/K K G L/K  L ) K inclusion F with ( ) ′ . L/K L/K , Gal ′ −→ ∼ ′ yn vrapiein prime a over lying m ′ = ) ( ) p eafiie bla extension, abelian finite, a be with ( S ′ Gal L ) L/K ) K ( ′ m ⇔ ) L/K K ( otiigaltoethat those all containing = ) L/K S ) and ( m , S Nm ) ) K ⊇ n tdfie an defines it and ′ L/K S sth , sth , ( p J K L S of S ) ⊆ Then . scon- is K K not C ′ ⊆ m F Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. I bla xeso.Ti sgaate yteeitnetheorem. existence the by guaranteed is This extension. abelian L hoe 5.1.5 Theorem field hr xssafiieaeinextension abelian finite a exists there aoia smrhs lim isomorphism canonical ta utet hsi h dl ls ru,wihi h anobjec main the is which th group, group class another idele the introduce is to this natural theory. cohomology quotient; more a be as to it out turns it directly, K m where , oeta h hoe osntipythat imply not does Theorem the that Note h field The alasubgroup a Call hnw tt h following: the state we Then hoes5 Theorems eoe h ru of group the denotes of H S hnetenm ftesubject. the of name the whence , ( m L ) ′ . orsodn oacnrec subgroup congruence a to corresponding 1 otistepie of primes the contains . n 5 and 4 ( Existence H of ←− S J H . m ( 1 K m m . C hwta,fraynme field number any for that, show 5 = -dasin )-ideals a m ) J . i ogunesbru modulo subgroup congruence K m → ( K o vr ogunesubgroup congruence every For ⊇ Gal m , H 1 ) L/K 42 ( ⊇ · L K K Nm i yn vrapiein prime a over lying ab ( uhthat such , and K /K L/K m , .Rte hnsuyn lim studying than Rather ). 1 J ) ( L K m J L m h ru of group the a vnasnl nontrivial single a even has ) H scle the called is S S ( K H m ( m m ). hr sa is there , if , ) modulo ′ iel in -ideals ←− four of t thas at class m C m m , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. hstm a enuigte ntewogwy—adi afa orIhad I 2009). hour Mathematicians an MIT half in and sa — way t suddenly wrong idea the I in the now them had and using I been worked, beginning had never I very they time the wen this but I from So fields, see, again.” You cyclotomic Law discouraging, Reciprocity garden. the ye was the the sp prove three It in nothing to for had down try Law. 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Reciprocity the about story a you tell will I 43 di h Gen- the it ed aytries. many e tl,I Still, ue. htall that w ti,and it, at u and out t etrue. be agree o ca to ecial use o and ars ad it. h Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ehdt banterm n entosi lblcasfil hoyfrom theory field class global principle, theory in field local-global definitions th class and the and cal theorems theory, the applying obtain number in for to local lies mean method and significance suitable global its a between and represents transition divisors se a of two modigication permits precisely slight first it a more the is or in ideles ideals, of Chevalley, notion of by The intoduced [6],[8]. first extensively use were which [6] Lang lztoswihwr eesr ntecasclieltertctre theoretic ideal classical the series Dirichlet disappeared. in subsequently ie necessary plethor methods, were a analytic which to The led alizations has results. and reaching transparent far particularly is methods cohomological K a felements of rosaes icl u ttesm ieterslsaeo su of are results the time 1936 same in Herbrand the nowhere J. at perhaps power. but great is difficult such there so distinction: are a proofs make to necessary ento 6.1.1. Definition If scle the called is ntefloigw ilconsider will we classes following Idele the In and Ideles 6.1 theory field class Global 6 Chapter where an , p a 1 Let ls edter a euainfrbigdffiut hc i which difficult, being for reputation a has theory field Class o lotalprimes all almost for ( = a S p K a p ∈ ustruhalfiiest fpie of primes of sets finite all through runs ) ea leri ubrfil.An field. number algebraic an be K ∈ a ru fS-ideles of group p p ∗ I K ∈ san is , K a p ∗ p Let 1 seta component essential uhthat such ∈ h eeomn ftegoa hoyuigiee with ideles using theory global the of development The . K S I K S p p ∗ eafiiesto rmsof primes of set finite a be hnthe then , . = I p Y K ∈ p S h union The . = agsoe l rmsof primes all over ranges K ideles [ p ∗ S a × I p 44 K S p Y r the are ∈ / ⊆ si ekrh[] 3,Mle[]and [4] Milne [3], [1], Neukirch in as S U Y p p of idele ⊆ K a oa components local Y p ∗ if K p a a sthe is , K p K of cec hoyi hc the which in theory a science p ∗ sntaunit. a not is h group The . hpretsmlct n of and simplicity perfect ch K atyjsie.Bti is it But justified. partly s K safamily a is but , dl group idele n hi gener- their and a ftenotion the of p tethave atment fteidele the of saui in unit a is hc sa is which toswe ctions atthat fact a erefore ( = of of a a K lo- p ) . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Proof. of h eopsto ru of group decomposition the Let em 6.1.3. Lemma ento 6.1.2. Definition where C by nta fteusual the of instead map K N y h diagram The 4 3 commutes. If 2 enwwn odfieteieecasgopof group class idele the define to want now We If 1 = x v K L N ∈ (( I := | epoete4 the prove we K Y w K L L/K K ,.,x .., x, L | ( /K v ι ∗ x {z ( Q N ≤ x ecos lc i prime) (ie place a choose we , = ) ) w ∗ K L L sGli,then Galois, is od ow s h olwn definition following the use we so do To . | } ) w v v as ) x ( N ≤ { x [ ( w L K rpriso h ommap norm the of Properties = ) L : M K w v : Let p ( ] w th x , σ then , | ( w ∈ v F ∀ .,x ..., L/K rpry suigmr pcfial that specifically more assuming property, G/G } ) Y x interchangeably) = G ∈ w w w w { N I N 0 1 0 x σw eGli with Galois be K N 0 ... , K N M = K (where o any For . L ( = K L 0 ( = L = { I x σ , σw v σ L ∗ x  = ) N ι N w 0 ∈ ∈ K ( L ) L N M x N N 45 ,w v, G Q G 7→ K L = ) : K K L L w v · } I .,x ..., σ : 1 N v L ( y ∈ σw = .,y ..., / / σ K K Gal n xaplace a fix and I L r rms.w s hs symbols these use we primes.. are ( = → x ∈ K ( G {  w ∗ 0 ι G/G Y σw ∈ ( n I v y = L/K = K ... , ... , v L  0 ) w N w Gal ∗ σ , ) 0 = ) = ) K ) L 0 eoti the obtain we τ σx } w v ∈ ∈ n ( Gal L/K x y G/G K ( Y L stefco group factor the as w σw ) w edfieanorm a define we , 0 0 0 | v /K } eobtain we , v L/K ) v τx -component = sGalois. is Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Proof. (because hrfr a eomitted) be can therefore em 6.1.4. Lemma hc mle that implies which or x Reminder: tefil ntels rce sabs ed ec nain under invariant hence field, base a is bracket last the in field (the ( = σ Let ∈ p x G/G N ≤ Y Y v O ) x ( ψ v k K 7→ w ( = = ) K Y 0 k ( = | ψ τ p N x h olwn iga commutes diagram following The ∈ Q p p σ w K σGal L f f ∈ k p ) k ( G/G | w ≤ Y | x p p O )= )) v ( N ⇒ ∈ L k K ( Y w σw K σ L x I 0 p v ∈ k L (( 0 v τ ) p G/G ψ Y Since . v p /K ∈ ( ,.,x .., x, = p K ( N N G x ( v p (( w N K k w Y ( L ) ) ≤ 0 Y k σ 0 Y K k I N ( )= )) L J O ◦ − σ | L/K L p k p )=( = )) L  1 L K ι L ψ τ | ( ( τx N i hswyw e rma dl oa ideal). an to Idele an from get we way this (in N x L ∈ ( x N ψ ) k Gal K f L = ) N N K K L L )= )) τσ L k = k 46 p K K w v L L | ( N p ( ( 0 ( k Y ( |{z} x − v L σ ) ( x ι K L / / k σ v k ∈ x w 1 k J I ◦ ( f k )) − ( x ) 0 G/G K Y K k (  k k {z x /K 1 N x ) ) ψ p | ) ) = x p ) k ... , K K ) . N .., , v ) L v archimedean w k ∈ σ ) = ( Y 0 ( ∈ x τσ Q x τ G ) N k ∈ K K ) − L σx Gal K L v 1 ∗ ( |{z} ( x x } k ( = )) Y ≤ L Y σw O N − L 0 places K ( L /K k ( ) x v v k ) ) ( = ) στσ x k ) = ) − 1 x σ and , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. eoeby Denote aea h lsr of closure the as same ≥ oolr 6.1.5. Corollary nieecasgop n e htteidcsaeeul Let equal. are indices the that see i and on t groups based methods), class inequalities analytic idele with index inequality on norm index the norm second from the deduce proved group we section class this In idele Generalized 6.2 group. class idele an Let h lmnsof elements the ed o n valuation any for field, h aia da ntecmlt oa ring local complete the in ideal maximal the oain6.2.1. Notation subgroup eo)i ico etr1i the in 1 center of disc a is below) to h lmnsaeuisa l places all at units are elements the where oain6.2.2. Notation .Hence 0. m o all For ecnue” use can we A nindex an ; P eteitga lsr of closure integral the be U U stegopo rnia das h etaedfie below.. defined are rest the ideals, principal of group the is v U U ( v v A m v ( v m ( ( o v eset we := ) m m := ) h lsr of closure the isi h lsr of closure the in lies := ) P ) c o nta f” of instead ” m ≤ aea have C O W , U nidx” index An ( h omidcsamapping a induces norm The ∗ K ( v v ∗ U + 1 ( v ∗ R R o l o rhmda n complex and archimedean v for m all for := m m v A ∗ > ∗ sfollows: as I , v ) = ti aldtern of ring the called is It . m p 0 U ≤ o h field the (of ai bouevalue absolute -adic m v v m O 0 = (0) K m m v ∗ N A en pieto ”prime means m v J ∗ K L 0 = ≥ in sacce n lorfrt 6 o oedetails. more for [6] to refer also and cycle, a as ” ( m m scle h ugopof subgroup the called is m : Z 1      K | I ) p 0 = ≥ L ste”ru ffatoa das ”prime ideals” fractional of ”group the is ” in ai field. -adic v R O C v C {z A /L 1 n let and , L ∗ ∗ v ∗ dividing n ec h lsr of closure the hence and , o rhmda n real and archimedean v for K 47 K ∗ } etern fagbacitgr of integers algebraic of ring the ie , → n n integer any and ) nonarchimedean v real v complex v o v for I | K o C o {z m m ≤ /K v K = emysythat say may We . p = n r even are and eas hi resat orders their because 1 and ai neesin integers -adic ∗ } A p p nonarchimedean etelclrn at ring local the be ≡ m ( 1 uisin -units m mod K/ ∈ ≡ Q N m o eoe based ones he K 0 ( 1 eanumber a be ) in i locally (ie ” W edfiea define we v K o v mod K and el (we deals m v ∗ ( v v p sthe is (see ) m p All . m v are K v ) is , . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. hs opnnsa all at components whose eak6.2.3. Remark em 6.2.4. Lemma tall at Proof. since element • • e now Let • efidthat find we 2) )w have we 3) ⇒ since ⇒ ⇒ Let W W I K W m α α v places edfietemapping the define we m m (1 m x − − α m := ( φ 1 1 ( = := = v − x x ( ∈ ≤ := ) | α { v K ∈ m α x K Q − I x I ∈ {z v − m m 1 ∗ v I ( = = . ∗ v x 1 m ) U U | x φ v ossso l ideles all of consists } m ∩ ) ehave We uhthat such v Q v 1) v x = ( ∈ W ( ssurjective. is I ) m m v v m < ) I α m I v I v v v m m h map the o all for ) m − ( ) eabtay yteapoiaintermteei an is there theorem approximation the By arbitrary. be v v ∈ /K 1 v ≤ ) ( ≤ K x ǫ α v I K eaccein cycle a be × K K m eas m because ) v | I m Q ( m v ∗ o all for ∗ : = ∼ α stesbru ossigo l ideles all of consisting subgroup the is = φ x v Y v = v are φ | − ∤ I/K v m m : K | xK ∈ sijcie.adw rv surjectivity. prove we and injective.. is I xK x W m W ∗ m v m W v m m ∗ ) /K if ∗ ∩ m ( v v ( | x ǫ < m v -units. v ǫ (= ) m I 48 7→ 0 = m ( K |{z} ( = ) m ( O v scoe ucetysmall sufficiently chosen is dl ls group class idele × ) → edfietefloigsubgroups: following the define we , o all for v ∗ xK U x o l v all for v Y v o v for v ∤ ( I/K ( m m K ) ∗ v K v ∗ )) hs opnnsare components whose v v ∗ ∗ ֒ → ≤ ∤ | m m I I | K ) m ≤ } ) I K = x m ( = v -units x v ) v Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ⇒ ujciiyof surjectivity scle h gnrlzd dl ls ru tahdt h cycle the to attached group class idele (generalized) the called is rpsto 6.2.6. Proposition ls group class ento 6.2.5. Definition Proof. oeinjectivity. fore eobtain we ψ since eobtain we since ⇒ ⇒ ⇒ since o n cycle any For Let (ie ψ hsw banamapping a obtain we Thus ( K ( α ( sinjective. is xK osdrtemapping the consider α x x − m = ) − x ∈ ψ |{z} α 1 W ∈ m 1 K x − a rirr hssosthat shows this arbitrary was x aihson vanishes W m I m 1 = (1) = ) ψ m ) = ) m b , ( ψ b n suethat assume and x |{z} vni otie in contained is even ∈ ∈ x ( = ) simdaebecause immediate is I ψ v m m ker ( o some(any) for K h quotient The ∈ hr sa isomorphism an is there |{z} ∈ m ( ψ U α P dl hoei nepeaino eeaie ideal generalized of interpretation theoretic idele ) W ) m v ⊆ ( o some for ) m m ψ P v I eeet in (elements : m o all for ) m I ψ m /K C x − /K ψ m ψ 1 = m b ento of definition (by ( ( b : := x P m W ab ∈ I ) W m W m x v m I ∈ I = ) ψ | 49 m ∈ m m m → 7→ S α = ∼ P hsimplies This . /K sovosysretv. epoethere- prove we surjective.. obviously is ∞ K → W m ∈ J ( K J m x m m ( ( K m J ) W m m W ( W ( m r nt talplaces), all at units are ⇒ ) ) m m m /P m ) ( /P b m 1 ) (1) = ) P m m ) m in K . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. hsipisteasrino h rpsto.T rv hsequation this prove To Proposition. the implications of both assertion the implies this Then ψ rpsto 6.2.8. Proposition Proof. .. h omrsdegroup residue Let norm The 6.2.1 ento 6.2.7. Definition sion osdrtig locally things consider I − 1 ” eclaim we o l places all for Aim: yPooiini h rvossbeto earayko that know already we subsection previous the in Proposition By ( L/K L/K P ⊇ m ecnie h mapping the consider we ” = ) ea xeso fnme fields. number of extension an be dl hoei nepeaino h omrsdegroup residue norm the of interpretation theoretic Idele iff K m W v m I | | K m cycle A and . /K ( ihu K without Let n l places all and ψ − {z ∗ I 1 N m m ( K P K L /K dlsideals ideles m I m ea disbeccein cycle admissible an be m I ? m W J L N } = ∗ m K | W ) K m W L ≤ n hnisr K insert then and ψ Q ( m {z J m K m L : v ( ) v ∗ v I ( = ∼ /P w v } ) m m x m | ⊆ v J 50 )= )) m v → 7→ . ( P N N in m m J ( K L K ) L x N K /P ( K J J w v ) m K L L P ( L ∗ m ) m scle disbefrteexten- the for admissible called is ( J N m w ∗ m ∗ . L ) K L = ∼ ) ( m I L eetig r oeesymixed hopelessly are things here ) ∩ K | J I m o h extension the for K ( m ) /P m {z N K L J everify we L ( m L/K } ) . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 1 nterSo u eq. our of rHS the in ∈ hc mle that implies which ψ ψ obtain therefore we where A v hnw banthat obtain we then hc implies which Let ⇒ ⇒ eobtain we αN words other in enwwn opoethe prove to want now we ” Let ⇒ yapoiainTermteeis there Theorem approximation By O k ⊆ ( ( k ⊆ ( w ∗ αN | αN |{z} αN N N o all for 1 = ∈ ( A x N α ” P y L o all for K K L L k K ) k m ( ( {z ∈ ( ( βy α β y = ) m ∈ β ulidele full N )= )) K I − L ∈ − w m |{z} ψ 1 y K βy L 1 | αNy } ) K − v ∈ w ) ssffiinl ls o1 to close sufficiently is n suethat assume and ) J ⊆ {z k 1 I ∈ m | ( ψ L m ( m u ψ N } k P I αN ( ( I | o all for ) | and ) m m αN m m = m | ∈ ) J K where ( N ∩ ( {z L αNβ {z β I |{z} ( m ∈ ( u βy K β m L K m − I ψ ) m ∈ J ) 1 } ∗ } L − ) L ) − k ( J 1 ( ∈ α A ) 1 L ∤ m ψ ⊆ = ) m ( I ∈ poiedirection.. opposite )) ( ( m m | N α N ψ K x ,w eeta idele an select we ), ∈ ( ∈ {z ( u ( I x ∗ ∈ βy βy m ( ( = ) K and k ψ )= )) } ) ≤ ∗ − 51 ∈ 1 O n β and α ( y O β P ) L ( = v ∗ N m and ) ∈ K L N ( L y ≤ u K w L ∈ sth K ) J w L L v ∗ ∗ ( ∈ ) m ) βy A I ) then )), L w ( = ie ssffiinl ls to close sufficiently is A N α w ) w K L ∈ y I scontained is L sth Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. o yfloigtegieie fasrc ls edter nta w instead theory field class abstract of guidelines earlier. developed the following by so, ya element an by opoetermiig1 remaining the prove to esethat see we feeet,tu hr sa idele an is there thus elements, of oolr 6.2.11. Corollary Proof. eak6.2.10. Remark 6.2.9. Corollary Proof. fe hssalprnhssw otneorformulation: our continue we parenthesis small this After ohieulte ilspl swt h isomorphism the with us supply will inequalities both x obtain we Altogether hssosthat shows This Since Note: hrfr u qaini proven. is equation our Therefore loehrw obtain: we Altogether ( x = = ) [ C C αN K I K m ψ K /N ecnueteieetertcfruaino h omrsdegroup residue norm the of formulation theoretic idele the use can we K ( L : /K samsil for admissible is w x N ( ( = ) K L A ∈ K w L ∗ C ) N C I L · ∈ m K w L L α o n extention any For = ∼ Let I [ = ] I ) hence = S N L ( ∞ I K L = ∼ J αN ∈ |{z} m K ic furthermore Since . I K u α ( st K /K m J eamsil for admissible be m I ( = K w L omidxieult,w ilsv h ieo doing of time the save will we inequality, index norm ( ) K [ : I I m /P ( ∗ m K ∈ K /K A L/K ) α ) eas A because / /P m ) ∗ I ( ) : ( N m N ∗ N N x K m N N K L ∩ K K K L L = L B ∗ N eko htaleeet in elements all that know we K K L I L N N | J ( I I L K B αN I L L A ∈ L S L/K K K [ = ] L L L {z ∞ J ( k = ) 52 ( = ) K I I m ( =1 L = ∼ K L A L L = ( ) ∗ ] } m A ) J /K o all for C sth αN −→ ≤ thlsthat holds it W ∼ L/K αN ( n h li ssonabove. shown is claim the and ) K x · m w m [ ∗ /N K ∈ L L Gal N : ) ) K L ( = ∼ K I hnw have we Then . : L k |{z} K A AB L ∈ P m | K m ( ( I C ) I m B L/K L and K ] ∈ L N = ) ) /K I K ∈ L m ) J K ∗ w L N ( ∗ m K L N )] I K L L ≤ I L W [ L m : r norms are K ihwe hich ] Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. omlzdwt iia oiievle1 If 1. value positive minimal with normalized rvstefis seto.Tescn sanalogous. is second The assertion. first the proves nt rms thus primes, finite Proof. sacnnclhmmrhs from homomorphism canonical a as ihcomponents with rpsto 6.3.5. Proposition J -oue nelement onto An G-module: a eak6.3.4. Remark ru fiee hc aeuisa opnnsa l nt rms th primes, finite all at components as units have which ideles of group a itnto.Ti losu otikof think to us allows This distinction. n therefore and , rpsto 6.3.3. Proposition eak6.3.2. Remark on based 6.3.1. formulation Proposition the of Continuation 6.3 ′ K ∈ , If hr emninaanthat again mention we where P I K L for L L/K F Let r h ru fiel n rnia dasrespectively. ideals principal and ideals of group the are sa follows as is dlsadieeclasses idele and ideles hc eas eoeby denote also we which , F | p snra and normal is p . eafiiepieo h edKand K field the of prime finite a be I a h medn i netv ooopim of homomorphism) injective (ie embedding The K v F p /I ∈ a Let e easffiinl ag nt e fpie.Then primes. of set finite large sufficiently a be S Let Let K p S L ∞ ,s ehv h eldfie map defined well the have we so 0, = F ∗ L/K G stewl nw ru ffatoa daso K. of ideals fractional of group known well the is S σ h idele the I ∞ K = ∈ ( σa / a C G etesto nnt rmso and K of primes infinite of set the be enra and normal be ( F ′ φ G I K ) L/K I K I = S : F K K ensacnnclioopimfrom isomorphism canonical a defines ∞ p a = = /I I a , σa F σ 7→ = L · G p I eoe t aosgroup, Galois its denotes I K σa eew soit iha idele an with associate we Here . K S K S ∈ K 53 r rmsdntdas by also denoted primes are = I ∞ I p ∈ Y K ∗ K S ∤ · σ K ∞ ) I to = ∼ − K I · K = ∼ sasbru of subgroup a as p L 1 p K ∗ F J v ⊆ J /K ihcomponents with J K p ∗ ∈ K G a K a L p obviously , L ∗ /P ∈ F = F I K G K v L/K p ⇒ etevlainof valuation the be a t aosgop Then group. galois its p kerφ I ∈ L . U I L p = o lotall almost for scanonically is v I a , K S w ∞ en ∈ I which , without K S a L I ∞ K ∈ σ − K the I 1 in L F p Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. primes fKcnann n let and L containing K of . ooooyo h dl group idele the Let of Cohomology 6.4 of subgroups rpsto 6.4.1. Proposition ie, da na xeso field extension an in ideal nteterm twudb omc etr u vroehsgo has 19.9.1956” everyone Serre, but to a better, letter find much ”Grothendieck could so you be if would subject; it the Th on theorem, theory. paper the his group on in in himself theorem Artin mysterious) explaini by very ”Hauptidealsatz” (actually the on an paper to little a is Enclosed you. to responded: grateful very Serre be would I unramifie words, extension two abelian in largest explained the in principal become okn ihgroups with working where opsdit ietpouto ooooygop vrtelclfields local the over groups cohomology of product direct a into composed K dlsbhv differently. behave Ideles rnia dl nteextension the in idele principal rpsto 6.3.6. Proposition following: the with it prove We Proof. ∗ 2 σa Therefore . ehv h following: the have We ti elkonta nielo field a of ideal an that known well is It hc r loGmdlssneteautomorphisms the since G-modules also are which ntal ecnwrite can we Initially .t ysae aebe nbet n h crlay stat ”corollary” the find to unable been have I shame, my ...to L/K = h inclusion The F a above eafiienra xeso ihGli group Galois with extension normal finite a be o all for , I L L If . p ∗ . ∩ σ a I ∈ K K Let If H ∈ G ∗ q = L/K L 2 ( ⊆ F n because and ,I G, ∗ L K G npriua,if particular, In H ∩ L eapieof prime a be ∗ ihu en rnia da ntebs field base the in ideal principal a being without I = ∗ q I L L sa rirr nt xeso,then extension, finite arbitrary an is S hc implies which , ( K ∩ ecnseta hs a elclzd,e de- ,ie, localized be can these that see can we ) ,I G, G L = hntels rpsto hw that shows proposition last the then , I L K L/K ie, , U I ∗ p Y L L p L ∈ p p stiil Let trivial. is ∩ ) S = = I a = ∼ I eisGli ru.Then group. Galois its be 54 a K L P Y F Y ∈ F H ∈ | | p = × p L L q a L L U ∗ ( K p K Y ∗ F ∗ above G ∈ F ∈ / then , ∗ S L eee have even we F a eywl eoeaprincipal a become well very can I U L , ∗ K L ∩ P ghwtetermcnb reduced be can theorem the how ng L F ∗ sa dl of idele an is ttefiiepie.I tcnbe can it If primes. finite the at d p I ) eafiienra extension normal finite a be then , a K si ati h euto given reduction the is fact in is saraypicplin principal already is euiu ooooia proof cohomological beautiful ⊆ σ tc ni pt now. to up it on stuck t L ∈ ∗ n htaliel of ideals all that ing G G a ∩ ∈ nypruethe permute only I = K K ( L I G = K htbecomes that ∗ L/K ) K G and ∗ = a While . . ∈ I L K I L are G K K p K . . , . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. and hsioopimi ie ytecomposition the by given is isomorphism this I hpr’ Lemma Shapiro’s hoe 6.4.2. Theorem naie n ecnrfrt oa ls edter ooti h resu the obtain to theory field class local to refer can we and unramified Proof. idele o all for in aie in ramified and L p L oits to π G U hc hw that shows which Then hsioopimi ie ytecmoiino maps of composition the by given is isomorphism this then , π L F p a for sidcdb h aoia projection canonical the by induced is sidcdb h aoia projection canonical the by induced is q = I stedcmoiingopof group decomposition the is oits to . L p F I Q F = L | p cmoet If -component. σ L = Q ∈ then , F G/G -component σ S ∈ H S G/G F H H If q I ( σU L S ,I G, q q S F ( ( ie us gives ,I G, ,I G, F L I safiiestcnann l nt rmsof primes finite all containing set finite a is L p L σ ∗ H H F ) L L , p q H = ∼ H = q a ) ) ∀ p ( U ( F ,U G, p q −−→ −−→ ,I G, q −→ lim res res Q L p ( surmfidin unramified is ( ie , ∈ ,U G, / ,I G, σ H are S ,H S, ∈ L L S p H H H G/G q ) ) L I p L ( p q q a = ∼ = ∼ q L ,U G, ) G/G F ) ( ( ( G G 7→ = ∼ ,I G, F −→ = ∼ p H 55 over π Y q ∈ F F σL ( H q H L S ,U G, a p I , I , F L ( L 1 = ) F S H q G F ∗ q F ∗ L L p ) ( K nue -oue,tu applying thus G-modules, induced ( F G ) ) q G = ∼ L I ( p U , If . −→ −→ F G L F π π p 1 = ) L M L , U , F F −→ p π H H I L , 1 = ) hnteextension the then p F ∗ L F H q q ) ) safiie naie prime unramified finite, a is L F ∗ ( ( −→ π G G ) q F ∗ ( F F G L httksec dl in idele each takes that L , L , F F ∗ L , F ∗ F ∗ ) ) F ∗ ) hc ae each takes which K L hc are which F /K lt p is Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. rmsof primes a oa components local rpsto 6.4.3. Proposition .I dimensions In 1. oyl,adischmlg class cohomology its and cocycle, a c representing component a.Gvnachmlg class cohomology a Given way. group oolr 6.4.4. Corollary corollary following the yields If Proof. everywhere. t components its h bv rjcin a ahelement each map projections above The n nyi o vr component every for if only and o ic ehv nisomorphism. an have we since Now p F a ( ∈ σ hnw hnetefil,tkn oa opnnsi ffce codn t according affected is components local taking field, the change we When npriua h isomorphism the particular In o h attopoete ti ucett assume to sufficient is it properties two last the for also ∈ 1 H for for for .,σ ..., , I I H q K L H ( etitti ucint h group the to function this Restrict . G c c c 0 hnti smrhs ae h -ooooyclass 0-cohomology the takes isomorphism this then 0 N ( ∈ ∈ ∈ F q ( ,I G, a fteidele the of ) G L , , p c H H H hsi ucino h group the on function a is This . L F ∈ F ∗ L q q q L , , a .W a e bv hteach that above see can We ). ( ( ( = ) K G G G K p > q F ∗ hc a ecmue as computed be can which , L/K N/K N/L p = ) then , ∗ nidele An I c stenr fan of norm the is L h map the 0 G p Let I , hc eas ftedrc u r lotaleulto equal all almost are sum direct the of because which , I , K I , /N N L p N ∗ H I a G K /N ) ) K ) ( ( , q I σ cor ( ( /N L q ( G 1 inf ⊆ ,I G, a omTermfrideles for Theorem Norm res .,σ ..., , ≥ F = K G ∈ L L c N c 1 I a I L L F ∗ ) L 7→ c K p I c ) c ⊆ hsw have we thus , p K q ) ) /N = ∼ ∈ ∈ = ∼ c p .Tersligfnto from function resulting The ). = F c a p N p = N stenr fa idele an of norm the is M = M 56 H G X ∈ fadol if only and if 1 = F p a edsrbdi h olwn simple following the in described be can p G b I | inf enra xesosof extensions normal be q res p H F L F ( K H cor ,I G, L q ∈ c N L p ( F ∗ ∗ q G F /N ∈ ( F . L K G L ′ ( F G a F ∗ p ( c G hoeacocycle a choose ) H c F L , p G c ( p p ( F c hc ae ausi h idele the in values takes which L , ) F suiul eemndb its by determined uniquely is = q F ) F F ∗ L ( n aethe take and ) | ,I G, F ∗ sthe is ) p a F ∗ ) i ffi salclnorm local a is it iff ,ie ) p · N L N a oits to ) | G p K F p ,ie 1, = L cmoetof -component normal. F ∗ a b · F p ∈ K N -components -components a G a G I and ( ∈ L F I σ L 1 to N ffeach iff .,σ ..., , = G F L I ′ a | F ∗ L F c . to q | is o if p ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Proof. Br 6.4.8. Remark where h aei n elcsfrteocrn fields occuring the for replaces one if same the hr h nlsosaegvnb h netv (because injective the by given are inclusions the where oolr 6.4.6. Corollary hoe 6.4.5 Theorem esosi h oa hoyi lydi h lblcs ytecycl the un by the case of global role the The in played law. is reciprocity theory local the the prove in to tensions easy relatively is it groups the think to us allows of ooooygroups cohomology ihtepoet that: property the with utpiaiegroup multiplicative hoe 6.4.7. Theorem maps. inflation em 6.4.9. Lemma unity. of roots adjoining by formed extensions field K ( e’ now Let’s hsipista h extensions the that implies This • hsfrom Thus esaluetefloigrslsfo oa ls edtheory.. field class local from results following the use shall We • u eoeta esaethe: state we that before But K n aua ubr hnteeeit ylcccooi field cyclotomic cyclic a exists there Then number. natural a m and , [ m = ) L L/K eol ietepoffor proof the give only we | F [ L : F S K agsoe l ylcccooi extensions. cyclotomic cyclic all over ranges : L/K p K 2 = ] prove H p Let ] H 1 o l finite all for , nlclcasfil hoyw aese htteBae group Brauer the that seen have we theory field class local In ( ( e ylcetnin hc r otie nafil hc is which field a in contained are which extensions cyclic ie, , Hilbert-Noether Let o l real-infinite all for , G 2 H H ( K F G L 6 2 2 L , . ( L/K H K ( ∗ Br 4 eafiieagbacnme field, number algebraic finite a be G G . . F ∗ 2 7 Ω L/K eafiieagbacnme ed Then field. number algebraic finite a be ( = ) ( /K L , G K Ω = ) ∗ I , L , H /K H ) Ω 1 ∗ fa of 2 I , H H = ) / cyclic L/K ( ) ( H G G Ω 1 1 fteurmfidextensions unramified the of p L ( 2 ( = ) L/K [ ,L G, ,I G, / cyclic L/K ∈ ( F p L/K ) G /K ai ubrfield number -adic . 57 S Ω [ [ I , L p L /K ∗ L , 1 = ) 1 = ) L H p H wrt steeeet of elements the as ) I , F ∗ 2 ∈ 2 ( Ω 1, = ) ( H G S G ,tepoffor proof the ), I L/K 2 L . L/K ( L G omafil omto.This formation. field a form L/K L , I , h dl group idele the ∀ L ∗ F ) I , ) S eddc the deduce we , K L nt e fprimes of set finite a H ) steuino the of union the is 1 ( Br G L/K ic L/K ( K cyclotomic aie ex- ramified sexactly is ) o which for , I , I L L 1) = ) ythe by L/K Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. nesadtemi eut(u o h roso course!) of proofs the not results(but main the understand I oolr 6.5.2. Corollary h oeo h utpiaiegopo edi h oa hoyi take is theory local theory the field in class field global the a in of group group group class multiplicative Class idele the the Idele of the role The of Cohomology 6.5 nt omlextensions normal finite scci fodr[ order of cyclic is let c htteei aoia eirct smrhs ewe h abeli the between isomorphism group reciprocity Galois canonical the a of is there that ubrfilsadtenr eiu group residue norm the and fields number omto ihrsett h dl ls group class idele the to respect with formation L/K G [ quotient hoe 6.5.1. Theorem p n ewl hwthat show will we and seat tsffie oso that show to suffices it exact, is u qain n6 in equations our inv o l primes all for rms n [ and primes, C ∈ r o qa o1 ytepeiu em hr sacci yltmcfie cyclotomic cyclic a is there Lemma previous the By 1. to equal not are K = 3 N S rmti eoti sa as obtain we this From Let npriua ewl aet rv that prove to have will we particular In nwa olw efi omlextension normal a fix we follows what In hsflosimdaeyfo h following the from immediately follows This o h ateult od,because holds, equality last the Now aebe eiwn itecasfil hoy fwihIfin I which of theory, field class little a reviewing been have I S : F G fw omtecompositum the form we If . with ete(nt)sto primes of set (finite) the be ′ N | L/K | K H c G p ∈ C 0 c m 1 ( [ p L fpieorder prime of L H ,C G, ] → H | F L [ > 2 : L K ( 2 F H F F G ( p p L : G ] . : Ω ) 2 L K of o l primes all for 0 = . h dl ls group class idele The | L ( K /K 4 G [ = ′ p . : /K ,6 2, p G o h real-infinite the for 2 = ] L L/K K I , o h finite the for ] C c I , = Ω ]. ⇔ . K isi h group the in lies L/K say ) 4 I , L p G . ′ h : ,w have we 3, The . L ) inv L/K ( N ) C n H and G → fa leri ubrfield number algebraic an of L c N C = ) F ∈ fanra extension normal a of res H L rtfnaetlinequality fundamental first ′ | = ] p H L 2 N F p | | L ( of p 2 H H 2 G ( c ( ∈ ( = res | res 58 of G H G 1 0 N/K K .Btb oa ls edter and theory field class local by But 1. = C ( ( S L L L/K ,C G, ,C G, 2 K L L c L ′ H ( ′ o hc h oa components local the which for n [ and p m /K c F ,C G, C I , · saHrrn ouewt Herbrand with module Herbrand a is H 2 c ⇔ 1 = L I , K and 1 = ( p N I , L L 1 G hnw have we then , [ = ) p /N ) ) ( L ) L/K c L L L | | ,C G, L/K C ) p [ ∈ −−−→ ′ ⇔ L ) F res = G ,let ), L ⊆ | F S . : C = : ( p K I , L . H L K L 3 res L rtedek etrt Serre to letter Grothendieck, - ihacci aosgroup Galois cyclic a with p hsorami oshow to is aim our Thus . p n that and 1 = ) L F nohrwrs htthe that words, other in , m p 2 ] H lyhv h mrsinthat impression the have ally | · .Snetesequence the Since ). m o h elinfinite real the for 2 = ] ( o all for 1 = L L/K | G : 2 [ H c L ( N/K eteodrof order the be ) K G F 1 F K ( N/L p : ,C G, = ] ffiiealgebraic finite of I , osiueaclass a constitute K · res I , N p inv o h finite the for ] L sterelation the is ) N L p ) N | F ) ∈ H > F c p ′ anization S 2 | p K ( 1 = . ,C G, p c c c by n p p and L ld of = ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ltl,i eopssit exactly into decomposes ie pletely, H gi oa ls edter odtrieteorder the determine to theory field class local again If L ntesneta hntoo hs ebadqoinsaedfie,th defined, are quotients Herbrand holds. these equality of and two third when the that sense the in where aigueo h isomorphism the of use making in Proof. ∗ 1 q S ( hc r nt o l h primes the all for units are which h quotient the compute To ic [ Since • • oeta yPooiin6 Proposition by that Note • • eas f6 of because terms.. two the compute we nfc ehave we fact In • Hence • e’ o rv hoe 6 Theorem prove now Let’s | G ete obtain then We . H ( h bv smrhs meitl yields immediately isomorphism above the 1 = n N 2) 3) n 1) I F 0 L L 1 S L , Let ( h ubro rmsin primes of number the S I I S = ) h ubro rmsof primes of number the G h ubro rmsin primes of number the L K F ∗ = F L | otisalifiiepie n l rmsrmfidin ramified primes all and primes infinite all contains H .If 1. = ) = S L , = L : 0 | | I eafiiesto rmsof primes of set finite a be F ∗ I ( ∗ K H H L S ,I G, K S ) ∩ . | a rm ere rm of prime a degree, prime has ] 0 1 4 · ( ( · I = . L h optto of computation the 2 ,I G, ,I G, L K S L S ∗ H ( ) stegopo -nt,i h ru faltoeeeet in elements those all of group the ie S-units, of group the is ∗ | 0 1 if p q L L S S = ( ) ) G ehave we 0 = | | p h prime the H ehv odtrietefcos ed hsby this do We factors. the determine to have we , F n C L , h 1 q p L n since and , ( ( ,I G, F ∗ C sinert is = = ) . . L 3 5 = ) . S I . L uhaset a such 3 S L 1, S L S G ) hc i above lie which , hc r nr in inert are which , p · p = ∼ F h L rmsof primes yn under lying ( F ( H othat so , ∗ p 59 I Y eas G because /L ∈ L H S h of 0 S K ) ( ( 1 ∗ ,I G, I H · L ( L S h uhthat such ,I G, = ∼ q salclqeto.Let question. local a is ) ( hc ontleaoeteprimes the above lie not do which ( S L L I S G L L S S L ) S K etil xss hnw have we Then exists. certainly F /L ) thus , ehave we 1 = ) F L , = ∼ − F hti o nr piscom- splits inert not is that = 1 S F ∗ splits S Q | ) H G p H L ) N 0 ∈ ( S 1 G ( = H ( ,I G, F becauseG L , n 0 ( 1 G L F ∗ S + ) F because 1 = ) L | L , p . · F ∗ ( F n use and ) n 1) = − ns is so en n 1 ). Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. K extensions cyclic hw ob qa to equal be to shown oolr 6.5.3. Corollary em 6.5.4. Lemma K N L scntutdsc htiseeet r ersne ynr ide norm by represented are elements its that such constructed is okaon hsb osdrn nta of instead considering by this around work n let and ieetfo h aeo dl rus hr yteNr hoe o ide for Theorem Norm the by where groups, idele of groups case the from different primes. of obtain r ersne yanr dl,adteeoelein lie therefore and idele, norm a by represented are stegopof group the is naie in unramified hoe 6.5.5. Theorem n nyif only and field G = L otisthe contains a nntl aypie hc r nr in inert are which primes many infinitely has C enwprove now We hshstefollowing: the has This h o h computation the For Let ee2 a estse y6 by satisfied be can 2. Here n then and • • • esaluewtotpofaTermfo ooooyo ylcgop to groups cyclic of Cohomology from Theorem a proof without use shall We h iclyhr sta ecno roidcd hc dl clas idele which decide priori a cannot we that is here difficulty The sfiieygnrtdo rank of generated finitely is K ( K L 3 2 1 I n hc a h rpryta t ne ( index its that property the has which and , L ( S x I S L a p p otisthe contains = ) p K 0 otisaltepie above primes the all contains = eafiiepieof prime finite a be ∈ x ∈ = 0 x p K K ), I I h n ∈ K K ( S N 1 ( S x C S √ . ( p p 0 · = sanr fadol fi salclnr vrwee We everywhere. norm local a is it if only and if norm a is K Let uisof -units t ot fuiy nti case this In unity. of roots -th L x L/K fadol if only and if K ∈ h eodfnaetlinequality fundamental second the Let = ) 0 p ∗ I Let ), ∗ ) K K S p p h N p . t ot fuiy Then unity, of roots -th x ∗ ∩ ( L/K p sn this Using . fpiedge,mkn h diinlasmto that assumption additional the making degree, prime of esatwt h following: the with start We . L/K L 0 | . = H K h S ∈ [ C ( = ) K 0 ∗ K L K ( K eacci xeso fpiedegree prime of extension cyclic a be ,C G, (ie, S ( eacci xeso fpiepwrdge.Then degree. power prime of extension cyclic a be n ntl eeae frank of generated finitely and ∗ eko httegroup the that know we ) √ : p K Let . p ( N N x p x x L ( o yn vrtepienumber prime the over lying not G ) . 0 n − 3 ) , ∈ C − | . san is n t xdgop( group fixed its and 1 F n because 3 and 3 x S 1) [ = L U eoti h inequality the obtain we , ] − ∈ p eafiiesto rmsof primes of set finite a be 60 6 N C · p K S +1) K ( [ n l nnt rmsof primes infinite all and K C -unit) ∗ : K p / ∗ N eayKme xeso over extension Kummer any be N ( ) p p G : G − F C and 1) C L = ] L L L = saKme extension: Kummer a is . ] naxlaygroup auxiliary an 6 p p p N x C n 0 piscmltl in completely splits p G 1 K L − saui o lotall almost for unit a is C L S 1 L : S [ = C hsi completely is This . n ) F G K − L a culybe actually can ) = ∗ : 1. ∩ N p K K e hence le, suethe Assume . I G p K S L S Then . C uhthat such = of L e in ses F ] K S 6 which -units ∗ p N ∩ F p C L for K ⊆ K le is if S Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ltl in pletely [( ag oesr htteidx[ index the that ensure to large uaino [ of putation with a oss fntigbtnr dlss that so ideles norm but nothing of consist ahieeclass idele each dl group idele tension o which for [ ota ti osbet opt hsidx hs rprisaes are properties These index. this compute to group possible idele the is it that so osethat see To usle httecomponents the that ourselves hoyw have we theory S [ [( p opeeys that so completely ,adteeoeb 6 by therefore and 3, C L p I I ic h primes the Since . ∈ / F K K K S S ∈ • oehrwith Together talw st pi h optto f[ of computation the split to us allows It stiilysretv,adiskre ossspeieyo th of precisely consists kernel its and surjective, trivially is hsi refor true is This ota [ that so • ∗ ∗ S : : ∩ ehv [ have We neeetr aclto hw that shows calculation elementary An U n h rdc formula product the and , ∩ F K K p K ,w osdrtefloigdecomposition: following the consider we ], L p ⊆ ∗ F ∗ = ] ( : ) a L ( : ) /K p N set ; F F I [ I ∈ I L K S K F S p K S F F p ⊆ ∗ ⊆ hr ( where , ∗ ( I ∗ ∩ /K K a K S ∩ : : r=1;ti stiilytu for true trivially is this 1); = or · N K S I ∗ p srpeetdb omidele norm a by represented is p ∗ F K S F K K ∗ G ) L [ : ∗ C hc so ueylclntr,adtecmuainof computation the and nature, local purely a of is which ] p L = ] ∗ ]=[ = )] I ecos diinlprimes additional m choose we F p F ∗ ∗ = hs lmnsrpeetteieecassof classes idele the represent elements whose p L K F . for ]wihue lblconsiderations. global uses which )] : fw o set now we If . 5 = ] F tsffie yteNr hoe o dlst convince to ideles for Theorem Norm the by suffices it , ∈ F ∈ . / S F : = each 4 p = F | p · K S S { ∪ p 2 Q K n K [ = ] ). ∈ p [ Y S ontleaoetepienumber prime the above lie not do K ∈ p because · p ∗ ∗ I q ∈ S S [ = ] K p 1 Q S ∗ a S ( : ( n ti refor true is it and . q .., , I ∗ i ntekre;i h dlsin ideles the ie kernel; the in lie C K p Q a [ ( : 7→ p K K S F p → K ∈ p ∈ ∗ ∗ p / K I p S ∗ ) ∩ ∈ fec idele each of p K Y · S m : p S p ∈ p S ∗ | ( : Y K K ∗ p F ∈ } a × ) S ∗ | | p S F oconstruct To . : p K 61 p ∗ ∗ p − sfiie n tms esml enough simple be must it and finite, is ] a = ] i | Y ]=[ = )] F /K surmfidin unramified is K 1 p p =1 m p ∗ ∈ = ] ) · p = ∗ / where p ( K ∗ p / [( F ( ;since ]; K F 2 ( Q K : I q K ∗ K C | · i p K · ∗ F S p ⊆ ∗ p K ) S K p × ∗ ∗ p ) p | ∗ · ) p a | p ∩ n N ∗ p p − a p K : | p ( : Y /K ∈ p / ∈ 1 ⊆ K G F ∈ S stenme fpie in primes of number the is ∗ S ∈ K q ,hne( hence 1, = / C /K ∗ notoprs h com- the parts, two into ] F ∗ p 1 ⊆ F ∗ N ( : ) L U ..q , S F then , S ocmueteindex the compute To . ∗ = tms esufficiently be must it , L r om rmteex- the from norms are p ∗ S ∗ ) L F = ] ehv oseiyan specify to have we ∗ p m F /K because h map the , ] q / = i ∈ ∩ / [( because , p F L F s ideles ose K S K F ∗ F ∩ ⊆ I ( ∗ htsltcom- split that p K )] √ S K ytelocal the By . rgrls of (regardless p , N ∗ x x | ∗ F p : G 0 0 ( : ) tse by atisfied | ,s that so ), tmust It . F C p ∈ q = ) L a for 1 = i K since , U ∈ splits S p p ∗ I 2 ) K by S n p ∗ . ] Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. T n h eodfnaetlieult spoe,poie htw c we that provided proved, is primes inequality the fundamental second the and olw meitl rmti.Teinclusion The this. from immediately follows ( oolr 6.5.7. Corollary following the tdo rank of ated ulma6.5.6. Sublemma where and the tains opeeyadall and completely group htevery that ltl in pletely N opeeyin completely K p = ∈ ∗ loehrw hrfr have therefore we Altogether nfc h eie equality desired the fact In n ehv rvnTerm(..) rmtepeiu hoesw have we Theorems previous the From (6.5.5). Theorem proven have we and If sn 6 using ) S p K ( N G K ∩ K ( p ∗ K √ S = p = ) p ∗ p N x . S t ot fuiy n tesl olw ht[ that follows easily it and unity, of roots -th G 5 q p stegopof group the is = ) ∗ ∩ K . 1 [ n since , L/K ,w omlt hsa follows: as this formulate we 4, C . q .., , ( = ∈ L / + ( T F K √ p m K p S n aif h olwn condition: following the satisfy and K ∩ : ∈ x / m n suetefield the assume and ∗ − N S ) p Let K othat so S ∗ surmfidin unramified is x ( 1 G ∗ pitn in splitting saKme xeso over extension Kummer a is hr exist There 6= K ∗ U ) C K K ∈ p p n = L/K q ∗ . L ∗ ∗ 1 + ( and ∩ ] H . q .., , K K ∩ ∩ 6 m p x 0 S \ ∗ p p p ∗ ∈ ( \ \ eacci xeso fpiedegree prime of extension cyclic a be [ ∈ ∈ ) ∗ stenme fpie in primes of number the is ∩ ,C G, S ∈ C N n p uis eko htti ru sfiieygener- finitely is group this that know We -units. S S ( − For . K ( ( ( K ( p K K L Y 1 H K = ∈ n p : ∗ L p p r naie,then unramified, are S ∗ ∗ 1 ) nsc a that way a such in ∗ F ) − ) ) p ( ( K ) p p K ,C G, = ] = ∼ p ∩ p N 1 ( ∩ ∩ p n because and ∗ 62 √ H i \ p K ∈ ) / =1 m rmsof primes p p p p y6 by x \ \ ∈ ∈ L / / 2 n S × .B 6 By ). S S ( otisthe contains 1 = ) K − ,C G, ∗ ∗ ∗ m i q Y ∗ U U =1 m . i ehave we 5 · p p ∩ . [( L .HneteSbem yields Sublemma the Hence 4. K ( = ( = p ) F ⊇ \ ∈ q / ∗ . = ∼ K i S 5 ∩ K K K x × ∗ . stiil Let trivial. is , every 4 G K S S U ∈ m p q x nwihall which in ∗ ∗ Y p ∈ p N S / ∗ 1 ) ) K U t ot fuiy then unity, of roots -th S ( : ) . q .., , p p ∈ = = ∗ p ∗ .Moreover ). S = U U n ∗ for K K p p n ( : − p = ) − S ( ⊆ K p ,and 1, ∈ 1 ∗ √ p ) ∈ S / U ∈ p x / p S ∗ ] ihGalois with , p = ) S ) p S p piscom- splits x · ∗ = ] nchoose an ( , ∈ K htsplit that K K ∈ x S p S ∗ . p isin lies ∗ ) K n p con- split + so , ∗ m ∩ . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ooooysequence cohomology omi n nyi ti oal omeverywhere. norm a locally is Proof. it if only and if norm hoe 6.5.8. Theorem n oolr 6.5.9. Corollary Theorem extension L Proof. hnw have we then fdegree of xc sequence exact [ H eobtain we group omlsbru findex of subgroup normal H sas xc,w e that see we exact, also is K K ′ 1 1 stiil e sasm that assume us Let trivial. is 1 = ′ ⊆ = ( ( Assume o ylcetnin hoe 6 Theorem extensions cyclic For result: general more following the now prove We hsi ucst rv hsfra for this prove to suffices it Thus esethat see we G ,C g, : M K L p G 1 C , h euneof sequence The epoeti yidcino h order the on induction by this prove We ] · L p → ⊆ K n < n rmteeatsequence exact the from and 1 = ) L n < K H of H n therefore and 1 = ) ′ etoe earlier: mentioned L H biul [ Obviously . p ′ − 1 ehave we , , G 1 bandb donn primitive a adjoining by obtained ( H 1 fteorder the If . = 1 g G ( ( G H ,C G, 1 a re mle than smaller order has → L = ( n K 1 ′ ,C G, /K If nodrt eal oapy6 apply to able be to order In . ( 1 ′ fteextension the If G H ,C G, /K L → L/M L/K C , 1 ) L ( C , H → /,C G/g, H 1 = ) L L G ′ K o if Now . 1 K 1. = ) H 1 .O h te ad eas h sequence the because hand, other the On 1. = ) H mdls1 -modules ( sanra xeso ihGli group Galois with extension normal a is ′ ( p G ′ ) ,C G, 1 ; 0 n . ( : K −−→ ( g inf G M ,L G, K ′ H = /K L steGli ru fa nemdaefield intermediate an of group Galois the is ) L ] ′ 1 ) L /K −−→ K | 6 H C , ( inf G ∗ n < p ,C G, −−→ . L/K ) inf 5 | 1 C , p K . → ( sjs nte omo the of form another just is 8 snta not is H G − → ′ 63 H L p = ) L L n H H 1 -group ′ 1 hnb assumption by then , 1 1. = ) implies 1 = ) scci,te nelement an then cyclic, is ′ ( ota yteidcinhypothesis induction the by that so , L ( /K ,C G, 1 0 p < ,C G, L K ∗ ( ( H G ,I G, ′ C , → ′ 1 L p ( L = L ′ L G pwr hneach then -power, G p /K o vr extension every for 1 = ) I L ) ′ t oto nt to unity of root -th L L ) nti ae let case, this In . ) n −−→ n ′ C , res = ∼ −−→ → /K res n [ and ftegroup the of . L M 5 ′ C . ′ p C , H H ,w replace we 7, 1 = ) H L 1 1 L H L 1 ( ( → ′ ′ ,C g, ,C G, ( 0 ,adfo the from and 1, = ) G : ( G ilsteexact the yields 1 K L L ′ F H L /K ′ ) L , = ] 1. = ) 1 as Norm Hasse ′ ( G p F ∗ C , /,C G/g, Slwsub- -Sylow x g G h case The . ) p K Because . ∈ K L ⊆ = n set and , ′ K ) G ythe by G M ∗ L/K L/K ea be sa is = ) M , , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. G G ereeteivratmp n ihi h eirct a rmth from law group reciprocity the the relating it t with by way and map, canonical invariant a the the retrieve in law, isomorphism canonical invariant It a the referred. obtain construct we we that that properties sential compatibility the satisfies which h dl group idele the scci ftesm re s[ as order same the of cyclic is 6.5.10. Theorem Since ean ob hw sta o vr omlextension normal every for that is shown be to remains mle httecnnclhomomorphism canonical the that implies isomorphism et of nents ento 6.6.1. Definition ax b second explicitly, the map of this conditions compatibility know formation. the class to satisfies need it not that important do we however momorphisms, o vr prime every For sijcie hsi rcsl h seto fteHseNr Theore Norm Hasse the of assertion the precisely is this injective; is group u oli oso htteextensions the that show to is goal Our invariants Idele 6.6 formations. class of L/K L/K hr ( where h oa nain isomorphism invariant local The ihTerm6 Theorem With Let hnteodrof order the Then . G H eisGli ru.W aearaytedecomposition the already have We group. Galois its be L/K 2 c scyclic, is ( inv ,C G, ∈ F | p H L eanra xeso ffiieagbacnme ed,adlet and fields, number algebraic finite of extension normal a be ) F 2 L /K ( I nivrathmmrhs,a eurdb h eodAxiom second the by required as homomorphism, invariant an ) G p L p L/K of steudryn module. underlying the as inv H : . If 5 Let H H K − . K 0w aentytrahdorga oso that show to goal our reached yet not have we 10 I , L/K c H 1 2 2 p ( ( ( ehv rmlclcasfil hoyteisomorphism the theory field class local from have we ∗ L ,C G, H 2 L/K G G /N ∈ ( ) H c L/K G L 2 hnw set we then , ( ri eirct law reciprocity Artin H F = L/K 2 L/K G L /K ( 2 ,C G, L/K X ) I , L eanra xeso ihGli group Galois with extension normal a be ( p p G L C , = ∼ L L , : ∗ L ) inv K C , L F L F ∗ → H = ∼ ) /K otegroup the to ) .T hwti,w ilascaewt the with associate will we this, show To ]. ) L L 1 sadvsro h degree the of divisor a is inv M → M 64 ) ( p F p ,C G, p L , /K → L L/K [ H L K F F ∗ p [ /K c F ) L 2 L p ∗ p where , ( 1 /N ,b hoe 6 Theorem by 1, = ) : G p : 1 ∈ K omacasfrain owhat so formation, class a form stecmoiino he ho- three of composition the is K L L [ p F L ] F ] /K Z /K Z : 1 / H / nacransnew will we sense certain a In . p K ( Z p Z L , F 2 L/K L ] ( | ⊆ F ∗ Z p G F ∗ ) / ) L/K [ Z r h oa compo- local the are L hr sa invariant an is there : 1 I , K [ L L so orees- course of is ] omdwith formed ) oa theory, local e Z : / K . Z m. 5 ] H . . ,which 8, o fa of ion 2 ti is it ut ( also o ,C G, G = L ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Proof. where sa nlso,s that so inclusion, an as under rpsto 6.6.2. Proposition eosmad.I atclrw bana nain homomorphism invariant an obtain we particular In summands. zero hn( then h attofrua eur nythat only require formulas two last The If eew s h ovnint nepe h nainmap inflation the interpret to convention the use we Here inv P inv where ytefnaetleuto fnme theory number of equation fundamental the By inv inv Note [ ial,for Finally, inv L c F F : F F N/K N/L N/L N/K N/L [ Let ∈ ′ K L ′ . xdpieof prime fixed a lotall almost : sa rirr rm of prime arbitrary an is F H ] F ( ( ( · ( c inv : c res res res ′ 2 cor P K = ( ∈ sa rirr rm of prime arbitrary an is G N/K p p L L L K inv H c N/K ] inv c c c · c ∈ = ) [ = ) = ) 2 = ) inv ( L/K c H G N I , = inv F L/K P P 2 N N If c L ′ inv ( c /K F p X G and ) F p H ′ L/K p : K H ,s httesmcnan nyfiieymn non- many finitely only contains sum the that so 1, = ( /K N/L I , N/L p inv K P 2 N 2 c inv ( ⊆ L p p ( ] G F G c : ,then ), N · X over I , c F [ = | p L L/K F p inv H N F L/K | p [ N = ′ F L ⊆ ustruhtepie of primes the through runs 2 /L [ L ′ tflosthat follows it ) L N ( F N/K /K I , P p G N I , F F : ): : for ( over L L/K K p p L : K for res c ) r omletnin ftefield the of extensions normal are c P K ) ] p p → N 65 · c ⊆ ]) p = L I , F H inv p [ = ] ∈ c | · H H p over L and ) X 2 for inv [ H F ) ( p L N/K 2 2 N/K G ( → ( = L F 2 G G inv N N/L H ( F F : G : F N/K P N/K c [ 2 K stepieof prime the is ′ K L L/K L and ( /K F I , enormal. be G p F ] : 1 ] inv I , /K I , p N N/K K · I , c inv ) N N p p p ] N L Z c ) ). = stepieof prime the is I , ) F p / N ′ ⊆ Z = /L N F ′ L ) H /K F inv then , ( 2 res p L ( L/K c G p yn under lying L N/K F c c p I , = ) K K N then , ) lying F ′ . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. epc oteieegroup idele the to respect bla aosgroup Galois abelian snta smrhs.T aeti o ehv ops rmteieegroup idele the from pass to have we so, this make To I isomorphism. an not is hence o ls omto,ecp o httehomomorphism the that for except formation, class a for components ento 6.6.3. Definition oa opnnsof components local fw eoeby denote we If eol edt oethat note to need only we lotall almost soitdwt h xc sequence sequ exact cohomology exact the the with From associated importance. central of is theorem following then hr tcnb hw htteclasses the that shown be can it where Since eldfie n sidpneto h re ffcossince factors of order the of independent is and defined well h ybl( symbol The fteclass the of em 6.6.4. Lemma sclaimed. as H L 0 ( oteieecasgroup class idele the to P o ahprime each For Since inv hncagn rmteieeivrat oteieecasinvariants class idele the to invariants idele the from changing When G L/K p a N/K P p saui o lotalprimes all almost for unit a is H χ F I , p | ( ( p 1 cor L (( ,L/K a, a ehv ( have we ( inv ) p a G If . p K ∈ L/K ) χ Let N L/K , c χ ∪ p K F = ) χ ( ′ δχ = ) h etito of restriction the ,L/K a, p I , /K ∗ p L/K ( Let ∈ inv hnw set we then , h ybl( symbol the , a G K p P ( p a ) n h nain apn r eae sfollows, as related are mapping invariant the and ) χ ) ,L/K a, X L ( ,i olw htteextensions the that follows it 1, = ) p L/K ∪ ( ( cor p p (resp. F L , L/K a G ea bla extension, abelian an be I /K inv δχ p a L = ) L/K χ L , C F K 1 p p p : n h dl homomorphism idele the and /K p L N ∈ → · ( H F = ) hc eawy osdra ugopof subgroup a as consider always we which c a enwitouetefloigsymbol: following the introduce now We . F δχ inv ea bla xeso.If extension. abelian an be = ) /K F (( H ′ p p 2 L = ) /K L , o lotall almost for 1 = ) ( a p 2 G ∗ L/K Y p ( ) ( H p p ,τ σ, ) F G ∪ → L/K a ( P /K ∈ 1 ( χ cor p L/K δχ a (( ( L , I ) 66 p G G p to L I , p (( a ) K L , P (respectively L/K = ) L F .Thus ). ) I , a → L p G c F /K ∪ p F F ) ) /K L L ) p | /K n since and δχ p C → ) Q/Z , X F p ∪ = p inv p L /K ensa lmn ftelocal the of element an defines ) ) p δχ ⊆ [ → ) L inv ∈ p N χ a ∈ p G ) n by and : F 1 ( [ ) 1 ∈ L ,L/K a, ′ L L/K G K then /L ∈ F p a I L/K : 1 ] /K hstels rdc is product last the thus , L K F Z H K · c F δχ / p and F 2 ] /K Z inv (( ( Z/Z ( = ) a = ( G a ,τ σ, p p a ) L L/K = ) ( inv p L/K a F G surmfidfor unramified is inv ∈ ∪ /K = ) ) sa2-cocycle a is ) L/K N/L a δχ h conditions the p I L/K p L , K a aif with satisfy · p c · N ) N sabelian. is F ∗ ihlocal with (( L ) L/K a F r the are ) /K G ∪ the , L/K I ence p L δχ L ∈ F ∗ ) , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ( hnpsigt h extension the to passing when oe re n tsffie oso h aihn f( of vanishing the show to suffices it and order power olw aiyfo h eairo h oa omrsdesmo ( symbol residue norm local the of behavior the from easily follows the rm ubr.Wt hsrdcinw oet h culcr ftep the of core actual the to come we reduction this With number). prime Q rtr,hnew a suethat assume may we hence erators, agsoe h rm ubr n h nnt rm over prime infinite the and numbers prime the over ranges h uoopim( automorphism The sinjective. is r ersne yccce ihvle ntepicplieegr idele principal the in values with cocycles by represented are Then that using see we Q ih( with hoe 6.6.5. Theorem c n ene oso ht( that show to need we and sbjcie hsec element each Thus bijective. is cor when case Proof. y6 by Q H tension. a, ∈ p p ic y6 by Since . 2 ( ( ( Q Q Let Now ew iwteeeet o elements the view we ie eueti neto othink to injection this use We Let ζ ζ H G Q c . ( ) ) 6 ζ / / p L 2 δχ a ∈ . ( ( Q Q ) .Hnet show to Hence 2. 0 esatwt h ipeosrainta tsffie ocnie the consider to suffices it that observation simple the with start We χ = ) ζ ζ G / / H p p Q ) L Q eagnrtro h ylccaatrgroup character cyclic the of generator a be sagnrtrof generator a is L/K / eaprimitive a be L , 2 en h extension the means surmfidfor unramified is for 1 = ) Q a ( K saccooi xeso,ie extension, cyclotomic a is G c p 0 ∗ · L , ,w a vnasm that assume even can we ), N/ N ∈ r h oa xesosascae with associated extensions local the are = . 4 L/ ∗ H Q . and ) hr xssacci yltmcextension cyclotomic cyclic a exists there 7 Q N , H Q 2 If inv ( L 1 G a ,L/ a, δχ and c ∗ ∗ ( and ) G L/K ∈ L/Q ∈ ∈ N H ∪ L/K Q H H l L Q 2 : n L , c sanra xeso of extension normal a is H ∗ ( 2 inv 0 H t oto nt;if unity; of root -th speieytersrcino ( of restriction the precisely is ) Now . ,L/ a, G = ( C , p sacci yltmcetninof extension cyclotomic cyclic a is ( inv 2 G ∗ H G 0 L/K ( 6= ) inv ( L/K G L L/K L/ 2 L G ⊆ L/K C htteidcdhomomorphism induced the that 1 = ) ( L/ l Q F G c Q L/ H / L , L/ n oal aie for ramified totally and H / = ) c Q L , R L , Q L/K ∈ Q Q 2 c , Q 2 ∗ ehv oso that show to have We . ( tsffie ocnie h case the consider to suffices it 0 = ( L , p Z ζ ( ∗ ) = ∗ (( ζ G H tteeoesffie oso that show to suffices therefore It . G with ) ) Q n aesTermimplies Theorem Tate’s and ) sgnrtdb ot fuiyo prime of unity of roots by generated is ) 67 → L , saprimitive a is a then , L/K ∗ 2 inv N/K ) ) p ( ∗ G ( H ∪ → L steieechmlg lse that classes cohomology idele the as ) ,L a, L/ N/K L , L/ δχ 2 N , ⊆ ( H a Q Q G inv ∗ = ) F ∈ sasbru of subgroup a as ) L , 2 ∗ c L/K Q / ( ) tefi ylcccooi ex- cyclotomis cyclic a is itself l Q = G Q L/K ( ∗ ⊆ ,w assume we 2, = χ ζ p a h form the has ) L/ ∗ Q inv o oero funity of root some for ) I , 1. = ) ( rm6 From . χ H ,L/Q a, Q c ( L l containing a, 2 n L , G N/ 0 = ) ( t oto nt ( unity of root -th Q G L/ Q ∗ Q ( ) N/K ( p ( ζ Q a, ζ cor ) ) = ) = ) / p . Q / 6 Q I , Q Q = l . ( eobtain we 4 if ; h extension The . ζ L c H oup o hs gen- these for ) N ) ) Q L p 0 H / 1 ) ∞ / c then , a, nfc,if fact, In . Q p ( n 2 Q G ( = L ( to ) = Q of G L/ > ∗ with p . p L/K ( Q Q a ∞ .If 2. ζ L roof. , ) then , ) Q l K this ; / ∪ then I , Q c sa is / δχ Z L p ζ = ∈ p ) ) ). . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. where • r Q Proof. exact. is eto h oa omrsdesmo ( symbol residue norm local the of fect ojgto,dpnigo whether on depending conjugation, where obnn hs eobtain we these Combining power h homomorphism the ment ute hte rnti speieytekre,adi diinw addition 6.6.6. in Proposition and kernel, the precisely is it inv not or whether further ≡ p For 2 ( L/K eei biul ucst suethat assume to suffices obviously it Here h atTermsosta h group the that shows Theorem last The each for • sgna formula product the by but hrfr ( therefore • ζ u ) For 3 For 1 − c / p r 1 Q v • ∈ r p sasretv ooopim o h ylccs ehave: we case cyclic the For homomorphism. surjective a is p ≡ 1 p · Because . santrlnme hc sdetermined is which number natural a is oso that show To H = ic h eiu edof field residue the Since . Q stevlainon valuation the is → a p p 2 p a l − ( ( eoti,writing obtain, we , = 6= 6= H G a, 1 ∈ a, l · L/K p 2 p l Q Q Q , p ( ∞ v G v ( p p ∗ ( p ζ ( ζ ( , h uoopim( automorphism the L/K I , a ( 6= [ ) ) a L ) / inv H / ) a, L If · Q o p mod p Q : 1 with ) r 2 ∞ L , inv ) Q K ) ( L/K L/K ζ ≡ ζ G ehave we , ( ∗ ] = ζ = L/K L/K ) sgna ( + ) ( a, ( n / → a, Y a, : ζ inv Q . ( Z p sacci xeso,te h sequence the then extension, cyclic a is Q ew ae( have we ie , a, Q I , H Q = ) H ssretv,w suefis [ first assume we surjective, is Q ( generates p · p a, L/K Q L 2 p ( p 2 a ( Q ( ζ ( ) ζ ( p Q G ζ ) Q = G ( ) = ∼ and p / ) > a ζ / p L/K c 6= Q / L/K Q p ( Q ) u a, M l Q = ( ζ / 68 p p a, p p a, p · Q ) p Q ) v ) or 0 has ϕ I , / p ζ [ ) I , p ζ C p L Q Q p m ζ H ( [ ) L = ( a L / = L steFoeisatmrhs on automorphism Frobenius the is p ζ p a, = ζ : 1 ) ) 2 R = ( ) ) p H l ) ( K ϕ < a : = 1 ζ ζ v ζ a sete h dniyo complex or identity the either is ) / → G Q −−−−−→ ζ l K ) lmns clearly elements, inv u v p 2 Q ( ] = sgna / p ζ ( sitga.W osdrteef- the consider We integral. is L v a ( + · p Q ( ζ ] G p ) r F a L/K .Thus 0. [ ( Z p ζ ) nthe on ) · a L /K ) p Z L/K / v sgna ) / ζ a 1 = ) p Q euetedecomposition the use We . o p mod Z : 1 − p ( a 1. = ) K tsffie ofida ele- an find to suffices it , L , 1 ) L , [ · L Q = ] , F ∗ Z p ∗ u : 1 ) 6= l isi h enlof kernel the in lies ) / Q n K n l Z unit: a t ot funity of roots -th p p ytecongruence the by ] ehv oask to have We . v 1 Z p | L a ( / a | ϕζ : ) p Z · K r ehro not or hether 1 = → = saprime a is ] 0 ζ o l mod p thus , n ζ . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. nemdaefield intermediate olw aiyfo hs o every For this. from easily follows then H ylco rm oe ere tflosfo 6 from follows it degree, power prime of cyclic That noprilfato.B h rvoscs hr sa is there case previous the By fraction. partial into hsi eset we if Thus hc sietin inert is which n determine and with F then hrfr h re of order the Therefore eoti,uigthat using obtain, we that shows which inv group map the inv sa otteodrof order the most at is H 0 2 2 | p ( ( L/K L/K • G G 0 inv eko o that now know We L/K L/K inv n oa ls edter ilsa element an yields theory field class local and oso httegroup the that show To . eueasml ruetivligteodr fteegop.Since groups. these of orders the involving argument simple a use we L inv L/K I , C , F inv 0 L /K 1 L/K L hti eemndb h oa components local the by determined is that ) L/K .B 6 By ). → sas ujciei h eea ae[ case general the in surjective also is p 0 c inv H ssretv,w nyne oso htteodro h factor the of order the that show to need only we surjective, is yislclcomponents local its by = c L inv L L/K 2 = Since . i ( [ L G . fdge [ degree of L/K X 5 H F L/K c p . H inv [ 0 10 L 1 = H H 1 2 : c ( inv ( ssretv o n ylcextension. cyclic any for surjective is G : 1 K L H L , = X 2 G 2 i 1 K =1 p i ( L/K ( s p /K ..., L 2 L/K G 0 G → ∗ c 0 ] ( F 1 ) ] Z inv G L/K L/K siet,w ae[ have we , inert is n 1 /K c 1 · · · → + L C , L / i L/K , I , = Z i ∗ p = 1 L/K Z L I , L , H c : L , c → etedge f[ of degree the ie H p p 1 h xc ooooysequence cohomology exact the 1 = ) K inv s ) n C , = L i 2 /H 1 r c , ∗ = 1 2 ∈ c ( 1 ) = ] I isi h enlo h homomorphism the of kernel the in lies ) 69 1 = G ( i /H L p L [ L/K G + inv H L 0 = 2 iie [ divides ) L/K , → ( L/K p · · · 2 c 1 2 : 1 G .,s ..., , X i i r p ( L ( , c K =1 i s G L/K G C 1 i F I , osdrtedecomposition the Consider . ∈ p , L , 0 ] n = L/K L L/K s r 1 /K L p s . n + s H ... , 5 i r → ) L , ∗ i p hr biul xssacyclic a exists obviously there i . n p nfc qastekre of kernel the equals fact in ) that 3 2 Z → i r I , L , 0 i + L ( i L ∗ 1 o if Now . c G iie h re of order the divides ) L F p + L H ∗ Z : L 0 c 0 L ) c ) K i Z F : p 2 = = : 0 : ∈ ( /K K n eaedone. are we and ] G K K K [ ∈ n H 1 .Uigtesequence the Using ]. L p L/K p = ] L , otisaprime a contains 0 2 + H : 1 c [ = ] ( K G F ∗ 2 Z C , steeeetin element the is ( .Since ). L n ] G i L + /K L L = ) F Z 0 : I , p /K K 1 r L 1 i p where ] L/K with ) · · · 0 L , F ∗ p p 0 s r is 0 s ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. n ic y6 by since and o ehia esn ti ett let to best is it reasons technical field For the enlarge to have h olwn observations: following the n ocnie h union the consider to and hoe 6.6.7. Theorem inv map If tutivrat o h lmnso h groups the of no elements we the section, previous for the invariants in struct invariants idele the law studied Having reciprocity The 6.7 surjective. is using H sasretv ooopimi eea.Ufruaeyti sntthe not is of this element every Unfortunately for order general. in homomorphism surjective a is hs etito to restriction whose oee ytegroups the by covered m nain map invariant If 2 K c hr sacci extension cyclic a is there ( K If o h olwn twudb eycnein fw ol hwthat show could we if convenient very be would it following the For ∈ G inv ⊆ L/K endaoetefollowing the above defined N/K H H L 1 2 L/K ( ( ⊆ I , G G sanra xeso,te eoti rmteeatsequence exact the from obtain we then extension, normal a is L/K L/K N N 1 ssretv ntecci ae hsw banfrteivratmap invariant the for obtain we thus case, cyclic the in surjective is ,w banahomomorphism a obtain we ), → r w omletnin of extensions normal two are . inv 6 C , C , . H h nain a a eetne from extended be can map invariant the 2 L/K L h homomorphism The L 2 = 1 = ) and ) ( G H fw aeit con htfrec oiieinteger positive each for that account into take we If . H inv L/K [ 2 L H ( 2 L inv inv G ( : 1 [ 2 L/K c L G L , ( L/K K yfrigtecmoiu ihacci extension. cyclic a with compositum the forming by 1 ∈ G H Ω K K : −→ 1 ] → δ ∗ L/K /K L/K H K c Z 3 ) : : I , ( / = H → 2 ] H H G L I , Z Z L ( I , ∗ 3 L/K G inv 2 2 ) Ω / ( H oigfo ylcextension cyclic from coming ( ( → with L Z G L/K = ) G G ⊆ 2 ) L/K L/K ob nteiaeo h nain a,we map, invariant the of image the in be to I , Ω Ω ( I L ⊆ 70 G H /K /K L I , [ L L L/K ag vralnra xesosof extensions normal all over range m H 2 c ,teeatchmlg sequence cohomology exact the 1, = ) → L , L I , I , ( H | ∈ G 2 ssc that such is ) [ Ω Ω ∗ ( L C I , K Ω 2 G ) [ ) ) ( L L /K : L → N/K G then , → → K → ) : 1 L/K H I , K −→ 1 Q Q ,w e that see we ], j 2 Ω 1 I , ] / / ( onie ihteinitial the with coincides ) Z H I , G Z Z N / L/K L 2 ) Z ( ) G c L/K C , = L jc H C , .W tr with start We ). hnw set we then , Q 2 L/K att con- to want w L ( / G ) Z L/K o the Now . salready is ae In case. I , inv L L/K to ) K Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. the where H in of sions ento 6.7.2. Definition simplicity for define hr ntels oml eol edt suethat assume to need only we formula last the in where hsi o rei eea,btsilapist h ylccase. cyclic 6.7.1. the Proposition to applies still but general, in true not is This es htw aearaymnind ihrsett h dl grou idele the to respect with mentioned, already have we that sense ujciewudb qiaett h group the to equivalent be would surjective omtswt h maps the with commutes epoedi iia a sw i tteedo h rvossection previous the of end the at did we as way similar a in proceed we h oue osmlf hnsw ila eoeadi eea o eve for general in and before maps as inflation will injective we the things interpret formation, simplify To module. the Proof. surjective. is H hsdfiiini needn ftecoc ftepreimage the of choice the of independent is definition This ehv enhsivrat0 fcus hsol ok fteelement the if works only this course Of 0. invariant has seen have we eas w uhpemgsdffrol ya lmn in element an by only differ preimages such two because 2 2 ( ( sicuin.Mr rcsl,ti en htw omtedrc li direct the form we that means this precisely, More inclusions. as Because e’ oeta h homomorphism the that note Let’s nodrt en nivratmpfrabtaynra extensions normal arbitrary for map invariant an define to order In H L/K G K L/K 2 L ( If L/K ⊆ K sbigebde in embedded being as ) L/K agsoe l nt omletnin of extensions normal finite all over ranges C , hnw have we then , L H L ⊆ ssbrusof subgroups as ) isi h mg ftehomomorphism the of image the in lies ) 1 scci,then cyclic, is N ( L/K ,teextensions the 1, = ) If H H L/K 2 2 H H inf ( ( H H G G q 2 2 ( 2 (Ω L/K L/K H sacci xeso,te h homomorphism the then extension, cyclic a is j L/K (Ω ( j and H L/K ◦ 3 /K ◦ /K ( 2 inf H I , I , res G (Ω = ) res lim = ) 2 L L L/K ) = ) (Ω N /K ) ) L −−→ inf −→ −→ eif ie ; 71 H j j = = /K −→ L , ehave we ) [ L q H H inf res ( ∗ i h nainmp.Tikn of Thinking maps. inflation the via ) L H G H ) 2 2 H L/K K ( ( 2 L N L/K H = ∼ 2 G G ( 2 ( N/K ◦ 3 ( L/K ◦ L/K L/K ⊆ H L/K ( j j C , G 1 L omafil omto nthe in formation field a form ( L/K C , C , G L ) ) ⊆ ) ) L/K L L K N L , j ) ) N/K nti ae ie case, this In . eve h groups the view We . L , ∗ r w omlexten- normal two are H ebigtrivial. being ie 1 = ) ∗ 2 1 = ) c ( snra.W can We normal. is G ∈ L/K H 2 L , ( G ∗ L/K mit p ,which ), yfield ry j . C L/K being I , L c L as ∈ ), , Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. qa degree equal L of Because hoe 6.7.3. Theorem oolr 6.7.4. Corollary m Proof. where h xc ooooysequence cohomology exact the inv h we hence restriction, with and inclusion) as here (interpreted h ere[ degree the in h eak aeaoe eko httemap the that know we above, made remarks the ec if Hence ssretv,s that so surjective, is H Thus nelement an oso hs eueteieeivrat.B 6 By invariants. idele the use we this, show To ′ hstermhstefollowing the has theorem this , 2 /K L H ( N/L ic o vr oiieinteger positive every for Since oso htteaoeieult si ata qaiyw osdrord consider we equality an fact in is inequality above the that show To egv o h olwn rca theorem crucial following the now give we efis hwthat show first We Therefore N/K and 2 eoe gi h edo l leri numbers. algebraic all of field the again denotes Ω ( res scci,te h extension the then cyclic, is | G ( f6 of H res N/L H K L .Bcueo h xc sequence exact the of Because ). 2 L inv ( 1 ′ c . 1 L L hnasml ru hoei ruetsosta fteextension the if that shows argument theoretic group simple a then , 7 ⊆ L c ( → N , c [ . N/L ′ L fadol if only and if 1 = L 3 n hshlside since indeed holds this and 0 = ) /K ∈ H : L ′ ′ H /K K ∗ H 2 : ⊆ ( .Since ). ) ( 2 ,hence ], K 1 res | L 2 ( If and 1 = ) N [ = ( G → ′ [ = ] N/K /K L/K L L c H r w omletnin.te ehv inclusions have we then extensions. normal two are H L c H ′ H /K [ = ) = ) H 2 ′ 2 L 2 ( 2 H ( ⊆ : sa lmn of element an is ) ( L 2 ( L , L/K H jc : res sanra xeso and extension normal a is L/K N/L K L (Ω 2 ′ H K L /K 2 ( and , ′ ′ [ = ] G ( /K ∗ /K H L 2 ] L : ) then , ( ) L c ) = ) K ′ 3 L/K → ′ scci,ti od y6 by holds this cyclic, is /K ⊆ ) ( = = ) → /K L G ] ⊆ c res H · H N/L res H L : m I , = ) H inv ∈ H ). ′ K / cyclic L/K 2 2 /K 2 L L ( 2 ( hr sacci extension cyclic a is there L ( 2 H G 72 N/K .O h te hand other the On ]. ′ ( c N/K L/K ( H ( ) N/K [ L , sas ylc o let Now cyclic. also is L/K L jc 2 −→ isi h enlof kernel the in lies 2 ( ′ j /K G ′ ( = ) ∗ c L/K ) H ) H ) L ) [ = .If ). I , ⊆ ′ ⊆ = ∼ 2 H 2 /K −−−→ jres res ( L . ( H 7 L L/K H L 2 ). ′ H I , . ( ) ′ L h homomorphism the 1 ′ N 2 L/K 2 /K 1 → L : (Ω L (Ω ( K c ′ j H G = fadol if only and if ) L ) /K ) /K H L ] 2 omtswt inflation with commutes ′ ⊆ ) L /K · ( ′ 2 /K N/L inv ) ( ) · H . L L 6 L , ylcetninof extension cyclic a ′ . 2 ′ L /K fadol if only and if 6 ( | ) stecompositum the is ′ H G ′ /K ∗ c v h formulas the ave N/K eobtain we 1 = ) ) j 2 ∈ ( c L/K → L/K n therefore and 0 = H I , 1 2 res N ( ) fdegree of L | .From ). L ′ divides /K c 1. = ers. ) ⊆ Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. a htteei nte element another is there that say eto h hieo h hieo h preimage the of choice the of choice the of dent Proof. surjective. is H hoe 6.7.5. Theorem hs etito otegroups the to restriction whose hsw banahomomorphism a obtain we Thus scmail ihiflto,i a eetne oacnnclhomomorphi canonical a to extended be can it inflation, with compatible is L/K enlo h mapping the of kernel ,c c, sacci extension cyclic a is ti h atsection.. last the in it ento 6.7.6. Definition in we section, previous the in seen have following we the as surjective is which H of lse.Fo h homomorphism the From classes. h map the define Because 2 2 H ′ ( ( fcus ehv ocnic usle htti ento sinde is definition this that ourselves convince to have we course Of ie hsterm ti ayt bancasivrat o h eleme the for invariants class obtain to easy is it theorem, this Given Let ssurjective, is G G ∈ hr emyasm htti xeso ss ag that large so is extension this that assume may we where , 2 L/K L/K (Ω H If K /K c 2 c I , L , ( ⊆ = G ∈ = ) L ∗ L/K L jc ) sw a e ydfiiinadti a nain .W proved We 0. invariant has this and definition by see can we as ) H → ⊆ = 2 S I , c (Ω N H L = h homomorphism The L jc /K 2 H ) If etonra xesos eas h map the Because extensions. normal two be ′ jc ( L/K hno course of then , ⊆ L/K 2 j c ,te tflosfo h rvostermta there that theorem previous the from follows it then ), ( o some for L/K : H ∈ inv H H H H H H 2 uhthat such .I hs r o ujcie hnw tl have still we then surjective, not are these If ). H ( 2 2 2 2 2 2 G K inv ( ( rmteivratmpo h dl cohomology idele the of map invariant the from ) ( ( ( ( 2 G G G G G G Ω (Ω : L/K N/K /K Ω Ω K L/K L/K H /K c H /K /K c I , 2 ∈ 2 = ( I , I , I , I , I , I , ) ( G Ω c c H G L o ucetylrenra extension normal large sufficiently a for ) N inv ′ Ω Ω L L Ω and c ∈ 73 ) L/K 2 ) ) ) ) /K ) ∈ ( and → H −→ −→ −→ −→ G −→ K j j j j j I , L/K c H c 2 I , H H H H H ( H Ω ∈ = c L/K 2 L 2 ) ′ ( 2 2 2 2 2 r h nta homomorphisms initial the are ) ( Q I , G ( ( (Ω (Ω → ( ie nyb neeeti the in element an by only differ L/K jc L/K L/K N/K / Ω L , /K /K Z .Snefracci extension cyclic a for Since ). /K Q ) c ⊆ / n hsb neeetof element an by thus and ) c I , ) ) ) ) Z ) ∈ H Ω ∈ H with ) 2 ( H 2 G ( 2 G Ω ( /K Ω G /K Ω c I , /K c I , Ω = ∈ atcm to come fact ). I , Ω ) H jc Ω hnwe then , .Let’s ). 2 ′ then . ( L/K pen- sm nts ), Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. eesr,bcuei eea h map the general in because necessary, rpsto 6.7.7. Proposition definition last the from obtain [ sntsretv.Hwvrfrteeeet nteiaeof image the in elements the for However surjective. not is h eoruigcci xesos hc ehv ecie by described have we which extensions, groups cyclic using detour The eas h reso h lmnsin elements the of orders the because r isomorphisms. are extension hoe 6.7.8. Theorem osqetyaempe oteol subgroup only the to mapped are consequently h etito of restriction the dl ooooyclass cohomology idele H extension and L 2 : ( rmtels ento eoti homomorphism a obtain we definition last the From ebifl ealtecntuto ftemap the of construction the recall briefly We If L/K K c ]. inv ∈ H ;i particular in ); 2 H L L/K L/K ( G ′ 2 /K ( Ω L/K c /K ilsahomomorphism a yields = feuldge [ degree equal of I , inv ,te eoti h invariant the obtain we then ), h nain maps invariant The inv Ω and ) If L K inv inv inv ′ c /K c c otegroup the to ∈ = L/K L/K L/K H ∈ c H H inv inv 2 jc = H 2 ( 2 inv G (Ω ,where ( : : : K K inv 2 G ( L/K H H H L /K L : : L L/K L 2 2 2 ′ ′ H H ′ /K ( ( ( /K ′ : I , L/K L/K L/K n nepeigifltosa nlsosis inclusions as inflations interpreting and ) /K 2 2 K c (Ω (Ω c I , L .I hscci aew aeb 6 by have we case cyclic this In ). 74 c H = [ = ] ∈ H ) L /K /K = ) ) ) −→ 2 ′ 2 H inv j with ) ( ( → → → X L/K L/K ) ) 2 L H p ( L/K → → L/K [ [ [ : 2 L L L inv ( K iietedge [ degree the divide ) Q Q L/K oigfo nt normal finite a from coming ) : : : 1 1 1 c c ota y6 by that so ] / / K K K L ) inv [ = Z Z F ′ , L ] ] ] /K c ) Z Z Z jc : 1 L/K / / / ∈ K p Z Z Z n eobtain we and c H ] p Z c 2 ∈ / ycosn cyclic a choosing by ( Z G [ L j L/K . of 7 eimmediately we : 1 . nrdcn the introducing 3 K Q I , H ] / Z L L 2 Z ) / ( then , L : Z forder of ′ K . /K 7 and ] . an 1 = ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. eso fdge [ degree of tension leri ubrfil,Ωtefil falagbacnmes and numbers, algebraic all of field the Ω field, number algebraic c inv Proof. c since and hoe 6.7.9. Theorem following the is constructions our H aosgopo Ω of group galois a H h nain a nrdcdi h definition the in introduced map invariant the l nt xesosof extensions finite all inv K of Proof. eso edof field tension ∈ ∈ 0 2 2 ∈ [ ehv ytels hoe h isomorphism the Theorem last the by have we , ( ( L L .enwcm otemi hoe fcasfil hoy Let theory. field class of theorem main the to come now ..we • • h ar( pair The That xo I Axiom xo II Axiom L C H L/K N/K H ′ ′ /K /K where L if inv Let : 1 [ 2 L 2 ( K tsffie ovrf that verify to suffices It o h ro ehv ovrf h axioms, the verify to have we proof the For o naporaefiienra extension normal finite appropriate an for ( N/K K c c L/K : 1 sadvsro h ere[ degree the of divisor a is ) N/K ] and ) inv = K Z K = ⊆ / σ ] inv L/K Z Z then ) : and , ) ⊆ a L ,C G, o vr omlextension normal every For : ∈ H / Set . ⊆ L K Z G ⊆ L 1 ′ /K hnb atscinteei a is there section last by then , H /K sbjcieflosnwesl rmtefc htteodrof order the that fact the from easily now follows bijective is 0 ( Ω L/K . h formation The N inv ⊆ 2 L sovosyafrainadtefnaetlrsl fall of result fundamental the and formation a obviously is ) res c 0 ( ′ efr h union the form We . N/K = c r w omletnin and extensions normal two are N inv L/K inv K : L o vr omlextension normal every for 1 = ) a = c 0 K etoetninfilsof fields extension two be ie , N/L Then . L/K ∈ respectively. ) [ = ] r enda h etitosof restrictions the as defined are jc H σ inv ( inv c res 2 : ∈ ( H = N/L L/K inv L N/K L C H 2 σ ( ( c Ω : ,C G, | L/K L/K 2 [ = ) L L ( .Frtepofo h formula the of proof the For ). ssurjective. is c K scnnclya canonically is c L 75 = : ∈ ′ ,s that so ], Ω /K K L ) sbjcie Let bijective. is inv ) C → n hrfr iio fteorder the of divisor a therefore and ] C : L sacasfrainwt epc to respect with formation class a is L/K = ) K Ω L/K ⊆ [ L ] = 6 · C . : 1 inv 7 c S H fec nt xeso edof field extension finite each of Ω K . 6 H K L/K 2 . ] N/K ( Z 2 C c L/K K ( G / K L c ∈ Z 0 mdl:If -module: c ′ eset we , with ∈ /K where L/K .Then ). L H H ′ = ) 2 /K inv 2 ( N/K G ( fec nt ex- finite each of L/K K K L G eacci ex- cyclic a be ′ H K /K usthrough runs to = 2 0 c ,then ), inv oml If normal. ( I , G H L/K ∈ eafixed a be L Ω L/K 2 C ′ /K ( with ) Ω N/K .If ). 0 say , c c the = ∈ ) Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. a omtost h aeo leri ubrfils fw gi eoeby denote again we If fields. number algebraic of case the to formations 6 invariants idele the for formula analogous the use we and 6.7.10. Theorem oolr 6.7.11. Corollary bijective. is extension the rmtefruai 6 in formula the From hoe 6.7.12. Theorem ilsacnnclioopim ethe ie isomorphism, canonical a yields ob nepee sicuin,w aeb 6 by have we inclusions, as interpreted be to ie yteformula the by mined tion ( fteieegroup idele the of c , inv eas fti hoe ecnnwapyteetr btatter fcl of theory abstract entire the apply now can we theorem this of Because rmti eimdaeyoti the obtain immediately we this From o h case the For h nes ftercpoiymap reciprocity the of inverse The ∈ udmna class fundamental L/K G H L/K ab N/L 2 ) ( M/K [ G : ( G L res fteGli group Galois the of C Ω L/K ab : K /K K L → c I , q ] containing = ) = ∼ · Ω = C G inv H with ) h ooopimcppoutwt h udmna class fundamental the with product cup homomorphism The h a u rdc ihtefnaetlclass fundamental the with product cup map The K L/K ab u − inv − L/K M/K . hsyields this 2 6 2 inv . ( M/L n sn h ovninta h nainmp are maps inflation the that convention the using and 2 θ G ftenra extension normal the of ihkernel with L/K ∪ c jc L/K L/K N [ = ( : H res = , H : , u 4 G u K L Z G ( q L/K c L L/K H L/K L/K ( ) : L/K ab jc G ⊆ hr ecnasm htteei normal a is there that assume can we where K −−−−→ 3 ri’ eirct law: Reciprocity Artin’s u L/K ( = ) L L/K ∈ L/K ] = N ) n h omrsdegroup residue norm the and · → θ eirct map reciprocity 76 ⊆ H L/K = ∼ inv L/K , [ inv ∪ L Z 2 N C 1 = ) χ ( ) M/K : H 1 K L/K C ( M/L ⊆ → G K sidcdfo h homomorphism the from induced is L /N 0 . 7 L/K ( the , ] M . L/K H jc 7 ( + L/K ) jres q L/K uhthat such ) [ = +2 Z hnw aetegeneral the have we then , omrsdesymbol residue norm C = ) ( L L L/K L c hc suiul deter- uniquely is which , = ) : C K . ewe h abelianiza- the between K 6 ) . ] inv /N .B 6 By 2. · c inv ∈ L/K M/L H N/K 2 C C ( ( res . L G K 7 c . M/K /N hr is there 5 L c L/K = ) I , . M C ass ). L Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 3 ure Neukirch. Juergen [3] K. Wingberg A., Schmidt J., Neukirch [2] 4 ..Milne. J.S. [4] 1 ure Neukirch. Juergen [1] Bibliography 6 eg Lang. Serge [6] 8 oci Mahnkopf. Joachim [8] Propp. Oron [7] Tate. John and Artin Emil [5] ebr 1999 delberg lxne cmd-pigrVra elnHiebr (2013) Heidelberg Berlin Schmidt-Springer-Verlag Alexander dto.Srne-elgBri,Hiebr,NwYr (2015) York New Heidelberg, Berlin, Springer-Verlag edition. -ii-class-field-theory-spring-2016/lecture-notes/ https://ocw.mit.edu/courses/mathematics/18-786-number-theory http://www.jmilne.org/math/CourseNotes/ ls il hoya h nvriyo ina S2016.] SS Vienna, of University the at Theory Field Class leri ubrter,Casfil theory field Class theory, number Algebraic leri ubrtheory number Algebraic etr oe ncasfil hoytuh ySmRsi tMIT at Raskin Sam by theory,taught field class on notes lecture lericeZhetere pigrVra elnHei- Berlin Springer-Verlag Zahlentheorie. Algebraische ls il hoy TeBn etrs dtdby Edited Lectures- Bonn -The Theory. Field Class Klassenkoerpererweiterungen ls il Theory Field Class 77 pigrVra e ok n.1994 Inc. York, New Springer-Verlag . ooooyo ubrfils Second fields. number of Cohomology ejmn e ok 1967 York, New Benjamin, . 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