http://dx.doi.org/10.1090/fim/021
FIELDS INSTITUT E MONOGRAPHS
THE FIELD S INSTITUT E FO R RESEARC H I N MATHEMATICA L SCIENCE S
Brauer Typ e Embedding Problems
Arne Lede t
American Mathematica l Societ y Providence, Rhode Islan d The Field s Institut e for Researc h i n Mathematical Science s
The Field s Institute i s a center fo r mathematica l researc h activity , locate d i n Toronto , Canada. Ou r missio n i s to provid e a supportiv e an d stimulatin g environmen t fo r math - ematics research , innovatio n an d education . Th e Institut e i s supporte d b y th e Ontari o Ministry o f Training, College s and Universities , the Natural Science s and Engineerin g Re - search Counci l o f Canada , an d seve n Ontari o universitie s (Carleton , McMaster , Ottawa , Toronto, Waterloo , Wester n Ontario , an d York) . I n additio n ther e ar e severa l affiliate d universities an d corporat e sponsor s i n both Canad a an d th e Unite d States . Fields Institut e Editoria l Board : Car l R . Rieh m (Managin g Editor) , Barbar a Le e Keyfitz (Directo r o f the Institute) , Thoma s S . Salisbur y (Deput y Director) , Joh n Blan d (Toronto), Kennet h R . Davidso n (Waterloo) , Joe l Feldma n (UBC) , R . Mar k Goresk y (Institute fo r Advance d Study , Princeton) , Camero n Stewar t (Waterloo) , Norik o Yu i (Queen's). The autho r wa s supporte d i n par t b y a Queen' s Universit y Advisor y Researc h Com - mittee Postdoctora l Fellowship . 2000 Mathematics Subject Classification. Primar y 12F12 , 16K50 ; Secondar y 16S35 , 12G05.
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Library o f Congres s Cataloging-in-Publicatio n Dat a Ledet, Arne , 1967 - Brauer typ e embeddin g problem s / Arn e Ledet . p. cm . (Field s Institut e monographs , ISS N 1069-5273 ; 21 ) Includes bibliographica l reference s an d index . ISBN 0-8218-3726- 5 (alk . paper ) 1. Inverse Galoi s theory . 2 . Brauer groups . I . Title . II . Series . QA247 + 200504281 3
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Introduction v Acknowledgments vii i
Chapter 1 . Galoi s Theor y 1 Notation 1 1.1. Integra l element s 1 1.2. Algebrai c extension s an d splittin g fields 2 1.3. Separabilit y 4 1.4. Galoi s extensions 6 1.5. Th e Galoi s group a s permutation grou p 9 1.6. Reductio n modul o prime s 1 1 1.7. Cyclotomi c fields 1 3 1.8. Cycli c extensions an d Hilber t Sat z 9 0 1 5 1.9. Radica l extension s an d solvabilit y 1 7 1.10. Th e norma l basi s theore m 1 9 Exercises 2 1
Chapter 2 . Invers e Galoi s Theor y an d Embeddin g Problem s 2 7 2.1. Invers e Galoi s theory 2 7 2.2. Definitio n o f Galoi s theoretical embeddin g problem s 2 9 2.3. Grou p cohomolog y 3 0 2.4. Braue r typ e embeddin g problem s 3 4 Exercises 4 0
Chapter 3 . Braue r Group s 4 3 3.1. Basi c fact s abou t algebra s 4 3 3.2. Simpl e Artinia n ring s 4 4 3.3. Finite-dimensiona l centra l simpl e algebra s 4 6 3.4. Th e Braue r grou p 5 0 3.5. Cycli c algebras 5 3 3.6. Braue r typ e embeddin g problem s revisite d 5 6 3.7. Extension s o f the dihedra l grou p 5 9 Exercises 6 2
Chapter 4 . Grou p Cohomolog y 6 7 4.1. Th e cohomologie s i n dimension s 0 and 1 6 7 4.2. Galoi s Twis t 6 8 4.3. Th e long-exac t cohomolog y sequenc e 7 0 4.4. Th e corestrictio n ma p 7 2 iv Content s
Exercises 7 3 Chapter 5 . Quadrati c Form s 7 7 5.1. Basi c theory o f quadratic form s 7 7 5.2. Cliffor d algebra s 8 1 5.3. Quadrati c form s an d non-abelia n cohomolog y 8 3 Exercises 8 5
Chapter 6 . Decomposin g th e Obstructio n 8 7 6.1. Decompositio n alon g a direct produc t 8 7 6.2. Orthogona l representation s 9 5 6.3. Projectiv e representation s 9 8 Exercises 10 2
Chapter 7 . Quadrati c Form s an d Embeddin g Problem s 10 5 7.1. Group s o f exponent 4 10 5 7.2. Witt' s Criterio n 10 8 7.3. Group s o f exponent 8 11 4 Exercises 12 1 Chapter 8 . Reducin g th e Embeddin g Proble m 12 3 8.1. Reductio n t o Brauer typ e embeddin g problem s 12 3 8.2. Examples : Th e holomorp h o f the quaternio n grou p 13 3 8.3. Embeddin g problem s wit h cycli c kernel o f order 4 14 1 Exercises 14 8 Appendix A . Pro-finit e Galoi s Theor y 15 1 Introduction 15 1 A.l. Th e separabl e closur e o f a field 15 1 A.2. Galoi s extension s 15 2 A.3. Pro-finit e group s 15 4 A.4. Pro-finit e Galoi s theory 15 6 A.5. Pro-finit e grou p cohomolog y 15 7 A.6. Pro-finit e cohomolog y an d th e Braue r grou p 15 8 Exercises 16 0 Bibliography 16 3
Index 167 Introduction
Consider a Galoi s theoretica l embeddin g problem , i.e. , th e questio n o f embeddin g a Galoi s extensio n M/K wit h Galoi s grou p G — Gal(M/K) int o a large r Galoi s extension F/K, suc h tha t th e Galoi s grou p Gal(F/K) i s isomorphi c t o a specifie d group E an d th e restrictio n ma p fro m Ga l (F/if) t o G correspond s t o a give n homomorphism n: E — > G. Ho w d o w e approac h suc h a problem ? Ho w d o w e determine whethe r suc h a n extensio n F/K exists ? Ho w d o w e find i t i f it does ? O r (preferably) ho w d o w e find al l o f them i f there ar e any ? The answer s t o thes e question s o f cours e depen d o n th e kin d o f embeddin g problem considered , bot h with respec t t o the nature o f the groups an d the nature o f the field. Fo r instance, i f G and E ar e cyclic , it takes nothing mor e than elementar y Galois theor y t o solv e th e proble m ove r a finite field, wherea s i t take s clas s field theory (o r a t leas t som e reasonably sophisticate d algebrai c numbe r theory ) t o solv e it ove r th e field o f rational numbers . The method s o f algebrai c numbe r theor y an d clas s field theor y hav e i n fac t been put t o good us e i n studying thes e kind s o f problems: I n the 1920's , Schol z [76 ] considered variou s solvabl e group s (mostl y o f smal l order ) ove r algebrai c numbe r fields, an d i n the 1930' s Schol z [77 ] and Reichardt [70 ] independently prove d that al l finite group s o f od d prim e powe r orde r coul d b e realise d a s Galoi s group s ove r an y algebraic numbe r field, i n both case s b y buildin g u p the extension s throug h solvin g embedding problem s 'along ' a composition series . Thi s approac h culminate d i n th e 1950's wit h Shafarevich' s resul t tha t al l solvabl e group s ar e Galoi s group s ove r al l algebraic numbe r fields; cf . [32 ] o r [66]. Also i n th e 1930's , Wit t [94 ] considere d group s o f prim e powe r orde r p n ove r fields o f characteristic p, and essentiall y prove d al l involved embeddin g problem s t o be triviall y solvable . W e wil l touch briefl y o n this i n Chapter 2 . I n the sam e paper , Witt solve d th e proble m o f embeddin g a biquadrati c extensio n int o a n extensio n with th e quaternio n grou p a s Galoi s group . W e wil l ge t considerabl e mileag e ou t of that resul t i n Chapte r 7 . It i s clear that a n embeddin g proble m i s in a sense a 'local' problem: W e shoul d be able to investigate i t full y usin g only the extension M/K an d the homomorphis m 7r: E — > G, withou t havin g t o conside r 'global ' structure s lik e separabl e closure s and absolut e Galoi s groups. 1 In 1932 , Braue r [7 ] introduce d a typ e o f embeddin g proble m fo r whic h th e necessary 'local ' informatio n i s readil y available : I n thes e so-calle d Brauer type embedding problems, wher e the kerne l o f TT can b e identifie d wit h a group o f roots o f
1This i s no t t o sa y tha t separabl e closure s an d absolut e Galoi s group s ar e no t usefu l fo r theoretical consideration s - section 6. 2 o f Chapte r 6 i s a goo d exampl e o f thi s bu t simpl y tha t they ten d t o b e unwieldy , i f onl y becaus e w e d o no t generall y kno w them . VI Introduction unity inside M,2 thi s information i s contained in the cohomology group H 2(G, M*) , or th e relativ e Braue r grou p Br(M/K), whicheve r on e prefers . Thi s i s no t onl y decidedly 'local' , bu t quit e convenient , sinc e bot h grou p cohomolog y an d Braue r group theor y ar e well-develope d disciplines , providin g ampl e tool s fo r studyin g embedding problems . We shoul d mentio n her e that Braue r wa s no t i n fac t intereste d i n the embed - ding problems. H e was - naturally enoug h - interested i n the Brauer group , an d hi s concern was the classification o f finite-dimensional centra l division algebras over the field K, an d the connectio n t o embedding problem s wa s a reduction, a s h e writes, from 'non-commutative ' t o 'commutative' algebra . W e will g o the other way , usin g the fac t tha t th e structure o f Brauer group s i s reasonably wel l understood . This monograph , then , i s abou t Braue r typ e embeddin g problems , primaril y the cas e where ker7 r ha s prime order , an d som e related embeddin g proble m types . This topi c bring s togethe r Galoi s theory , Braue r grou p theory , grou p cohomolog y and the theory o f quadratic forms , an d al l o f these subjects ar e covered i n the text , in Chapter s 1 , 3 , 4 an d 5 . (Chapte r 1 differ s fro m mos t introduction s t o Galoi s theory b y not containin g ver y man y examples ; o n the other hand , ther e ar e plent y of explicitl y give n Galoi s extension s i n late r chapters. ) I n addition , Appendi x A gives an introduction t o pro-finite Galoi s theory . Chapter 2 provide s th e set-up , i.e. , th e basi c results , th e definition s an d th e first examples . Al l don e i n a s elementar y a fashio n a s possible. 3 Mos t impor - tantly, i t introduce s the obstruction t o the embedding problem . Thi s i s an elemen t in H 2(G, M* ) expressin g the solvability (o r non-solvability) o f the embedding prob- lem. Th e existence o f obstructions i s what make s Brauer type embedding problem s nice. Chapter 6 deal s wit h th e proble m o f decomposin g th e obstructio n int o con - venient factors . Again , sectio n 6. 1 i s fairl y elementary , whil e sectio n 6. 2 relie s o n pro-finite cohomology . Thi s (an d th e subsequen t sectio n 6.3 ) is , however, th e onl y place i n the monograp h w e make rea l us e o f pro-finit e cohomology , an d i t ca n b e skipped withou t givin g i t a second thought , shoul d on e s o desire. Chapter 7 explores th e connectio n betwee n Braue r typ e embeddin g problem s and quadrati c forms . Specifically , i t provide s criteri a fo r solvabilit y fo r a numbe r of embedding problem s i n terms o f equivalence s o f quadratic forms , an d describe s how to find the solutions . Thi s include s the famou s resul t b y Witt: A bi-quadratic extension K{yJ~a, yJb)jK i n characteristic ^ 2 can be embedded i n a Qg-extension , if and onl y i f the quadrati c form s (a , 6, ab) and (1,1,1 ) ar e equivalen t (ove r K). The final chapter, Chapte r 8 , i s concerned with reducin g embedding problems . As i t happens , i n some case s embedding problem s tha t ar e no t o f Brauer typ e ca n be reduced to Brauer type embedding problems and solve d as such, thus extendin g the usefulness o f Brauer typ e embedding problems. I n particular, i f the embeddin g problem i s non-spli t wit h kerne l o f prim e orde r p (an d th e characteristi c o f th e involved fields i s not p), i t ca n b e s o reduced . Another kin d o f embedding proble m that ca n be reduced to Brauer typ e i s the case wher e ker7 r i s cycli c o f order 4 . Suc h a n embeddin g proble m reduce s t o tw o Brauer type embedding problems , an d th e results coverin g this reduction ar e give n in sectio n 8. 3 o f Chapter 8 .
2See sectio n 2. 4 i n Chapte r 2 fo r details . 3Or a t leas t reasonable . I t i s perfectl y possibl e t o describ e Qs-extension s withou t resortin g to Braue r groups , bu t i t i s hard t o motivat e th e arguments . Introduction vn
There ar e o f cours e othe r way s o f lookin g a t embeddin g problems ; cf . [55 ] and [32] , which als o contain s extensiv e references . Particularl y noteworth y i s th e study o f embedding problems over Hilbertian field s (a s defined i n Chapter 1) , where it can be shown that split-exac t embeddin g problems with abelian kernel are alway s solvable. (Th e proof , whil e a littl e tedious , i s no t particularl y deep , an d consist s mostly o f producin g so-calle d regular extension s wit h prescribe d abelia n Galoi s group. W e refer t o [19]. ) Mos t o f the groups w e will consider ca n be realise d easil y as Galoi s group s ove r Q (o r an y Hilbertia n field ) b y invokin g thi s result . Thi s realisation i s generally not , however , ver y explicit . Another commo n approac h i s to realis e th e finit e grou p G a s a Galoi s grou p over C(t) (a s mentioned i n Chapter 1 , this i s always possible) an d then attemp t t o descend t o Q(t), followe d b y specialisatio n t o Q . Mos t realisation s o f finit e simpl e groups (sporadi c groups , projectiv e specia l linea r groups , etc. ) ove r Q hav e bee n obtained i n this manner. Here , a good referenc e i s Malle & Matzat [55] . A powerful method , whic h w e will make som e use of , i s the mor e general appli - cation o f group cohomology . W e will use it a s it relates to the Brauer group , bu t i t can easil y b e employed independently . Here , a n importan t pape r i s Hoechsmann' s Zum Einbettungsproblem [30] , covering the basic s o f this approach . Also , Neukirc h [65] made great contributions here, using the cohomological description o f class fiel d theory t o conside r problem s simila r t o those o f Shafarevich . Vlll Introduction
Acknowledgments This monograph grew-in a slow and round-about way-out o f my Ph.D.-thesis (University o f Copenhagen, 1996) . Thus , thanks are due to those who (more or less voluntarily) go t involved i n that project . Firs t an d foremost, thi s mean s m y thesis advisor, Prof . Christian U . Jensen, who-quite apar t fro m bein g helpful concernin g the actua l mathematic s - spent a goo d dea l o f time an d effor t convincin g m e that Brauer typ e embeddin g problem s ar e interesting . A s i t happened , h e wa s right . Also, thanks to Prof. Moshe Jarden, Dr. Dan Haran and Dr. Ido Efrat fo r comments and suggestions regarding my work during my stay in Tel-Aviv in the spring o f 1995. Thanks to Shreeram Abhyankar fo r asking the questions that le d to Chapter 7, 4 and t o Noriko Yu i for suggestin g tha t I try making a monograph ou t o f my thesis and variou s papers . The wor k o f producin g th e manuscrip t wa s partl y supporte d b y a Queen' s University Advisor y Researc h Committe e Postdoctora l Fellowship . Finally, two other things should probably be mentioned her e as well: Th e man- uscript wa s written i n AMS-WFQ&, an d computer calculation s relating to examples were performed usin g Mapl e V.
Arne Lede t
4 Although, unfortunately , hi s questions ar e not actuall y answered there . Bibliography
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[94] E . Witt, Konstruktion von galoisschen Korpern der Charakteristik p zu vorgegebener Gruppe der Ordnung pf, J . Rein e Angew . Math . 17 4 (1936) , 237-245. Index
[c] (cohomolog y class) , 31, 32 A4 a s Galois group , 134 (M,G,c) (crosse d produc t algebra) , 47 Anisotropic, (M, cr, 6) (cycli c algebra) , 47 form, 78 K(n) (cyclotomi c field) , 13 vector, 78 [-:K] (degree) , 2 , 49, 15 5 Artin-Schreier, 41 f'(X) (forma l derivative) , 5 Artinian ring , 45
(ai,.. . , an) (diagona l form) , 77 Aut/<-(M) (automorphis m group) , 4 ~, 50 , 77 Automatic realisation , 102
[21] (equivalence clas s o f algebra), 51 AutQ8, 133 ~G (fixe d points) , 4 , 30, 6 7 1 B (G1-)t 31 , 68 [G:H] (index) , 155 2 (a/p) (Legendr e symbol) , 14 B (G,-), 3 2 Mx, 14 2 Br( —) (Brauer group) , 51 x Br ( — ) (m-torsion o f Brauer group) , 55 , 56 M /4, 142 m Brauer group , 51 jR* (multiplicativ e group) , 2 Brauer typ e embeddin g problem , 34 nx, 8 3 Brauer's Theorem , 30 2top (opposit e algebra) , 48 1 U - (orthogona l complement) , 78 C (fiel d o f complex numbers) , 9 u J L v (orthogona l vectors) , 78 C(q) (Cliffor d algebra) , 81 (a,b/K)p (p-cycli c algebra) , 55 Co(q) (eve n Cliffor d algebra) , 81 a = p b (p-equivalence), 56 CJ(g) (eve n Cliffor d group) , 82 K*/p (p- equivalence clas s group), 56 C*(q) (Cliffor d group) , 82 (a,b)p (clas s o f p-cyclic algebra) , 56 C
167 168 Index
Complement (i n semi-direc t product) , 6 7 Elementary symmetri c symbols , 1 0 Completely reducibl e module , 4 4 Embedding problem , 2 9 Composite o f fields, 8 Epimorphism, 2 9 Conjugation, 7 4 Equivalence, cor (corestriction) , 33 , 72, 7 5 of algebras , 5 0 Corestriction, 33 , 72 of crosse d homomorphisms , 6 8 in dimensio n one , 7 5 of extensions, 3 1 in dimensio n zero , 7 5 of quadratic forms , 7 7 Covering group , Etale algebra , 7 4
of A n, 9 7 Euler's Lemma , 1 4 of 5n, 9 7 Euler's parametrisatio n o f 503(IR) , 8 6 Crossed homomorphism , 16 , 25, 26, 30 , 6 8 Even Cliffor d algebra , 8 1 Crossed produc t algebra , 4 7 Even Cliffor d group , 8 2 CSA (centra l simpl e algebra) , 4 6 Expressing equivalence , 7 8 Cyclic algebra , 4 7 Extension o f group wit h module , 3 1 Cyclic extension , 1 5 Cyclic group , Fg (finit e field), 6 as Galoi s group , 3 7 Factor system , 3 1 obstruction, 5 7 Faddeyev, 13 6 of order 1 6 a s Galoi s group , 14 6 Field, 1 of order 8 as Galoi s group , 115 , 116 , 14 5 extension, 2 Cyclotomic field, 1 3 First cohomology , 6 8 First cohomolog y group , 3 1 dx (degree) , 8 1 First Kochendorffe r Theorem , 16 1 d(-) (discriminant) , 1 1 First Supplement , 1 4 D2n (dihedra l group) , 38 , 59 , 101 , 14 8 Fixed poin t field, 4 DC (centra l product) , 9 4 Fixed poin t group , 30 , 6 7 DxC (pull-back) , 5 9 Formal derivative , 5 DD (centra l product) , 10 7 Frobenius automorphism , 8 Decomposition field, 1 2 Frobenius' Theorem , 6 3 Decomposition group , 1 1 Frohlich's Theorem , 9 6 Dedekind Independenc e Theorem , 1 5 Fundamental Theore m o f Algebra , 9 Dedekind's Theorem , 1 2 Fundamental Theore m o f Galoi s Theory , 7 , Degree, 156 of CSA , 4 9 of field extension , 2 , 15 5 G-extension, 6 Density Theorem , 4 5 G-group, 6 7 Diagonal form , 7 7 G-homomorphism, 6 7 Different, 1 1 G-module, 3 0 Dihedral group , 38 , 59 , 101 , 14 8 Gal (Galoi s group) , 6 , 10 , 15 2 D4 a s Galoi s group , 3 8 Galois algebra , 3 4 obstruction, 5 8 Galois closure , 7 Dg a s Galoi s group , 116 , 11 8 Galois extension , 6 , 15 2 obstruction, 6 1 Galois group , 6 , 10 , 15 2 Dimension (o f algebra) , 4 3 Galois twist , 6 9 Dirichlet's Theorem , 1 5 of CSA , 6 9 Discrete module , 15 7 of quadratic form , 8 4 Discriminant, 1 1 Gauss' Quadrati c Reciprocit y Theorem , 1 4 of polynomial, 1 1 General linea r group , 2 6 of quadratic form , 7 8 GL(2,3) a s Galoi s group , 136f f of quadratic polynomial , 9 GL (genera l linea r group) , 26 , 4 9 Division algebra , 4 3 GL(2,3), 13 4 Domain, 1 Group extension , 2 9 extension, 1 integrally closed , 2 HI (Hamiltonian quaternions) , 54 , 6 3 Double Commutato r Theorem , 4 9 Hp3 (Heisenber g group) , 3 9 Dual space , 7 8 H°(G,-), 30 , 6 7 i/x(G,-), 31 , 68 Eisenstein's Criterion , 4 if2(G,-), 3 2 Index 169
Hamilton's Theorem , 8 6 Matn(—) (matri x ring) , 4 3 Hamilton-Cayley's Theorem , 4 3 Merkurjev's Theorem , 5 5 Hamiltonian quaternions , 54 , 6 3 Merkurjev-Souslin, 5 6 Hasse invarian t Min (minima l polynomial) , 3 of quadratic form , 8 5 Minimal polynomial , 3 , 4 3 Hasse-Witt invariant , 8 5 M/K-vector space , 26 , 6 8 Heisenberg group , 3 9 Modular group , 4 0 as Galoi s group , 3 9 of order 1 6 as Galoi s group , 119 , 12 0 obstruction, 5 8 obstruction, 9 2 of order 2 7 as Galoi s group , 13 1 Hilbert Sat z 90 , 1 6 N (norm) , 16 , 30, 44, 8 3 additive, 2 5 Newton powe r sum , 10 2 Hilbert's Irreducibilit y Theorem , 1 3 Newton-Puiseux' Theorem , 16 1 Hilbertian field, 1 2 Noether equations , 1 6 Hochschild-Serre sequence , 12 6 Noether's Problem , 1 3 Hoi (holomorph) , 13 3 Non-quadratic residue , 1 4 Holomorph, 13 3 Norm, of Qs a s Galoi s group , 139f f of field extension , 1 6 on algebra , 4 4 HolQ8, 13 4 Homogeneous element , 8 1 on Cliffor d group , 8 3 hw(q) (Hasse-Wit t invariant) , 8 5 on G-module , 3 0 Normal basis , 1 9 Is(R) (integra l closure) , 2 Normal Basi s Theorem , 2 0 Index (o f subgroup), 15 5 finite fields, 2 0 inf (inflation) , 3 3 Normalised facto r system , 7 4 Inflation, 3 3 Nrd^/x (reduce d norm) , 6 3 Inner product , 7 7 Numerical decomposition , 9 1 Integral closure, 2 O(q) (orthogona l group) , 7 7 element, 1 On(K) (orthogona l group) , 8 3 extension, 1 Obstruction (t o embedding problem) , 5 6 Invariant basis , 2 6 Opposite algebra , 4 8 Invariant Basi s Lemma , 2 6 Order (o f pro-finit e group) , 15 5 Inverse Galoi s theory , 2 7 Orthogonal, Irreducible module , 4 4 basis, 7 9 Isometry, 7 7 complement, 7 8 Isotropic form , 7 8 group, 7 7 Isotropy vector , 7 8 representation, 9 5 vectors, 7 8 3 K Principal involution , 8 3 S (supernatura l numbers) , 15 5 Pro-finite group , 15 4 s(—) (Schu r index) , 4 9 Projective genera l linea r group , 2 6 crp (Frobeniu s automorphism) , 8 Projective representations , 9 8 Sn (symmetri c group) , 1 0 Priifer group , 15 7 Sn (coverin g grou p o f S n), 9 7 Pull-back, 3 3 Sp-polynomial, 1 3 of order sixteen , 5 9 Scalar extension , 44 , 6 8 as Galoi s group , 6 2 Scalar par t (o f quaternion), 5 4 Purely inseparable , 2 1 Schur index , 4 9 Purely transcendental , 1 3 Schur's Lemma , 4 4 Second cohomolog y group , 3 2 Q (fiel d o f rational numbers) , 1 1 Second i-C-group , 5 5 QL (trac e form) , 9 7 Second KochendorfFe r Theorem , 129 , 16 1 Q n+i 2 (quaternio n group) , 59 , 91, 101, 14 8 Second Stiefel-Whitne y class , 8 5 QC (centra l product) , 11 1 Second Supplement , 2 4 QD2n (quasi-dihedra l group) , 59 , 14 8 Section (o f group extension) , 3 1 QQ (centra l product) , 10 9 Semi-direct product , 32 , 6 7 Quadratic equivalence , 5 6 of order sixteen , 5 9 Quadratic extension , 9 as Galoi s group, 6 2 Quadratic form , 7 7 Semi-linear action , 6 9 Quadratic independence , 5 6 Separable, 4 Quadratic Reciprocity , 1 4 Separable closure , 15 1 Quadratic residue , 1 4 Separably close d field, 15 1 Quadratic space , 7 7 Serre's Theorem , 9 7 Quasi-dihedral group , 59 , 14 8 Shafarevich, 2 7 QD& as Galoi s group , 113 , 11 4 Short-exact sequence , 2 9 obstruction, 6 0 Simple extension , 7 Quaternion algebra , 5 4 Simple ring , 4 5 Quaternion class , 5 5 Skew field , 1 Quaternion group , 14 8 Skolem-Noether's Theorem , 4 9 as Galoi s grou p i n certai n prim e charac - SL (specia l linea r group) , 9 9 teristics, 10 1 SL(2,3), 13 4 of order 8 as Galoi s group , 105 , 10 8 SO(q) (specia l orthogona l group) , 7 9 obstruction, 9 1 SO (K) (specia l orthogona l group) , 8 3 of order sixteen , 5 9 n Solution (t o embeddin g problem) , 2 9 as Galoi s group , 6 1 Solvability b y radicals , 1 8 R (fiel d o f real numbers) , 9 sp (spi n norm) , 8 3 Radical, 1 6 Special linea r group , 9 9 extension, 1 7 SL(2,3) a s Galoi s group , 134f f Re (scala r part) , 5 4 Special orthogona l group , 7 9 Reduced grou p extension , 3 0 Speiser's Theorem , 2 6 Reduced norm , 6 3 Spin(g) (spi n group) , 8 3 Reflection, 7 9 Spinn(K) (spi n group) , 8 3 Regular module , 7 3 Spin group , 8 3 Regular quadrati c form , 7 8 Spin norm , 8 3 Relative Braue r group , 5 1 Split exact , 2 9 Relative norm , 12 6 Splitting facto r system , 3 1 Relative trace , 7 5 Splitting field , 3 Representing a quadratic form , 7 8 for CSA , 4 8 Representing a n element , 8 0 Springer's Theorem , 8 0 res (restriction) , 33 , 5 1 Square class , 5 5 Restriction, Square root , 9 on Braue r groups , 5 1 Standard representation , 3 0 on cohomolog y groups , 3 3 Steinitz number , 15 5 Restriction-corestriction, 7 3 Subalgebra, 4 3 Root field , 3 Subextension, 7 Roots o f unity , 1 3 Supernatural number , 15 5 primitive, 1 3 Sylow's Theorems , 16 1 Index Symmetric, functions, 1 0 polynomials, 1 0 Symmetric group , 1 0 54 a s Galoi s group , 13 6 Symmetric Polynomia l Theorem , 1 0 T(-) (tenso r algebra) , 8 1 Tv(-) (reflection) , 7 9 Tensor algebra , 8 1 Theorem o f Accessory Irrationalities , 8 Topological group , 15 1 Torsion o f Brauer group , 5 6 Tr (trace) , 25 , 30, 7 5 Trace, of field extension , 2 5 on G-module , 3 0 Trace form , 9 7 Transcendental element , 2 , 4 3 Transitive subgroup , 1 0 Translation Theorem , 8 Trivial module , 3 0 Uniformised facto r system , 3 2 V4 (Klein Vierergruppe) , 3 8 Vec (vecto r part) , 5 4 Vector par t (o f quaternion), 5 4 Weak solutio n (t o embedding problem) , 3 4 Wedderburn's Nor m Criterion , 6 5 Wedderburn's Theorems , 15 , 46 Whaples' Theorem , 2 8 Witt invariant , 8 5 Witt vectors , 3 6 Witt's Cancellatio n Theorem , 8 0 Witt's Chai n Equivalenc e Theorem , 8 0 Witt's Theorem , 10 8 Z (rin g o f integers), 1 1 Z(-) (center) , 15 , 43 Z (Priife r group) , 15 7 l Z (G,-)t 31 , 6 8 Z2(G,-), 3 1 Zero-dimensional (topologica l space) , 15 2 0th cohomology , 6 7 This page intentionally left blank Titles i n Thi s Serie s 21 Arn e Ledet , Braue r typ e embeddin g problems , 200 5 20 Jame s W . Cogdell , Henr y H . Kim , an d M . Ra m Murty , Lecture s o n automorphi c L-functions, 200 4 19 Jerem y P . Spinrad , Efficien t grap h representations , 200 3 18 Olav i Nevanlinna , Meromorphi c function s an d linea r algebra , 200 3 17 Vital y I . Voloshin , Colorin g mixe d hypergraphs : theory, algorithm s an d applications , 200 2 16 Nea l Madras , Lecture s o n Mont e Carl o Methods , 200 2 15 Brad d Har t an d Matthe w Valeriote , Editors , Lecture s o n algebrai c mode l theory , 2002 14 Fran k de n Hollander , Larg e deviations , 200 0 13 B . V . Rajaram a Bhat , Georg e A . Elliott , an d Pete r A . 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Dritsche l an d James Rovnyak , an d M . A . Kaashoe k Peter Lancaster , Editor , Lecture s o n operato r theor y an d it s applications , 199 6 2 Victo r P . Snaith , Galoi s modul e structure , 199 4 1 Stephe n Wiggins , Globa l dynamics , phas e spac e transport , orbit s homoclini c t o resonances, an d applications , 199 3