Lectures on Complete Discrete Valuation Fields 1: Discrete Valuation Fields (1.1). Valuations. One can generalize the properties of the -adic valuation ¡£¢ and proceed to the concept of valuation. Let Γ be an additively written totally ordered abelian group. Add Γ ¥§¦ ¤ ¤¨¦ ¤ ¥ ¤ ¤ to a formal element + ¤ with the properties + , + + , + (+ ) = + , Γ Γ Γ ¤ ¤ ¥ © ¤ (+ ¤ ) + (+ ) = + , for each ; denote = + . Γ A map ¡ : with the properties ¤ ¡ ( ) = + = 0 ¡ ¡ ¡ ( ) = ( ) + ( ) ¡ ¡ ¡ ( + ) min( ( ) ( )) ¡ is said to be a valuation on ; in this case is said to be a valuation field. The map induces Γ Γ ¡ a homomorphism of to and its value group ( ) is a totally ordered subgroup of . ¡ ¡ If ¡ ( ) = 0 , then is called the trivial valuation. It is easy to show that ( 1) = 0, and if !¡ ¡ ( ) ( ), then ¡ "¡ # ¡ "¡ ¡ ¡ ( ) min( ( + ) ( )) min( ( ) ( )) = ( ); $ ¡ ¡ ¡ ¡ thus, if ¡ ( ) = ( ) then ( + ) = min( ( ) ( )). %'& £(©) ¡ * £(©+ ¡ , * & (1.2). Basic Objects. Let = : ( ) 0 , & = : ( ) 0 . Then %'& coincides with the set of non-invertible elements of %-& . Therefore, is a local ring with the * % ¡ & unique maximal ideal & ; is called the ring of integers (with respect to ), and the field .& %-&0/1* 2©)%'& = & is called the residue field, or residue class field. The image of an element .& in .& is denoted by , it is called the residue of in . The set of invertible elements of %'& 34& %-&56* is a multiplicative group = & , it is called the group of units. 798;:-:-<= $ , & Assume that char( ) = char( & ). Then char( ) = 0 and char( ) = 0. $ .& Proof. Suppose that char( ) = = 0. Then = 0 in and therefore in . Hence = char( .& ). <A:'BDCE8;FHGAIKJ0<LC£MN<PORQSGATAFU<LTNVWJ0<LC£MN<PORQSGAT+I1QS8;C£VLFE= (1.3). >@? ¡ 1. A valuation ¡ on is said to be discrete if the totally ordered group ( ) is isomorphic to the naturally ordered group X . L[ Z ¡£¢ Y For a prime and a non-zero integer Y let = ( ) be the maximal integer such that ¡£¢ ¡£¢ Y)/E\ ¡£¢ Y ]¡£¢ \ ¡£¢ ¤ divides Y . Extend to rational numbers putting ( ) = ( ) ( ); (0) = + . The -adic valuation ¡£¢ is a discrete valuation with the ring of integers Y Kbdc _ Y6\6©WX`a\ %'&^ = : is relatively prime to \ 1 QiFkjRlN8POm8onp<ACEMm<DORQiGLT+qQS8DCEVLF 2 e0fhg r ^ The residue field & is a finite field of order . s t t ¡ ¡ ¢ ¢ 2. Let = ( ). For an irreducible polynomial ( ) define ( u ) similarly to above. ¡ ¢ The map ( u ) is a discrete valuation with the ring of integers ( t ) y y %'& w.x t t ©+s t t t z ^ = : ( ) ( ) [ ] ( ) is relatively prime to ( ) y ( v ) x ( t ) t / t s t s and the residue field is s [ ] ( ) [ ] which is a finite algebraic extension of . s ¡P{ The field has another discrete valuation trivial on : ( ) = deg( ), its residue field x x is s . (1.4). Discrete valuations. A field is said to be a discrete valuation field if it admits a |}©~%'& nontrivial discrete valuation ¡ (see Example 1 in (1.3)). An element is said to be | ¡ a prime element (uniformizing element) if ¡ ( ) generates the group ( ). Without loss of X generality we shall often assume that the homomorphism ¡ : is surjective. 798;:-:-<= | Let be a discrete valuation field, and be a prime element. Then the ring of %'& |9N%'& integers %'& is a principal ideal ring, and every proper ideal of can be written as for \, * |p%'& %'& some 0. In particular, & = . The intersection of all proper ideals of is the zero ideal. |9; \©+X ©3 & Each element 2©+ can be uniquely written as for some and . % \ ¡ ©6m Proof. Let be a proper ideal of & . Then there exists = min ( ) : and hence |9N%'&§ |9N%'& |9N%-& |9o©) for some unit . It follows that and = . If belongs to %-& ¡ ¤ the intersection of all proper ideals |9A%'& in , then ( ) = + , i.e., = 0. ¡ |@©34& |9; (©34& |9 |95 Let \ = ( ). Then and = for . If 1 = 2, then ¡ Y ¡ ©3 & \ Y \ + ( 1) = + ( 2). As 1 2 , we deduce = , 1 = 2. (1.5). Completion. Completion of a discrete valuation field is an object which is easier to work with than with the original field. ¡ ¡ X Let be a field with a discrete valuation (as usual, ( ) = ). is a metric space & ( ) / with respect to the norm = (1 2) . So one can introduce the notion of a fundamental sequence: a sequence ( ) 0 of elements of is called a fundamental sequence if for every R \ ¡ Y6\6!\ real there is ( ) 0 such that ( ) for ( ). \9 If ( ) is a fundamental sequence then for every integer there is such that for all ccc ¡ 2 \ ¦§\ ¦ \4Y\9 we have ( ) . We can assume 1 2 . If for every there \9 \9 ¡ $ ¡ ¡ ¡ \!\9 is such that ( ) = ( +1), then ( ) and ( ) for , and 0 0 ¤ ¡ hence lim ¡ ( ) = + . Otherwise lim ( ) is finite. 798;:-:-<= The set of all fundamental sequences forms a ring with respect to componentwise addition and multiplication. The set of all fundamental sequences ( ) 0 with 0 as ¤ / \) + forms a maximal ideal of . The field is a discrete valuation field with its ¡ ¡ ¡ ¡ discrete valuation ¡ defined by (( )) = lim ( ) for a fundamental sequence ( ) 0. Proof. A sketch of the proof is as follows. It suffices to show that is a maximal ideal of \( ¤ . Let ( ) 0 be a fundamental sequence with 0 as + . Hence, there is an ¢ 1 $ \!~\ \! ~\ \!£\ \ 0 0 such that = 0 for 0. Put = 0 for 0 and = for 0. © Then ( ) 0 is a fundamental sequence and ( )( ) (1) + . Therefore, is maximal. QSFkjRlm8DOm8on<LC£MN<PORQSGAT+qpQi8;CEVAF g e0f 3 (1.6). A discrete valuation field is called a complete discrete valuation field if every ©W fundamental sequence ( ) 0 is convergent, i.e., there exists = lim with respect to ¡ ¡ ¡ ¡ ¡9 ¤ ¡ ¡ . A field with a discrete valuation is called a completion of if it is complete, = , ¡ ¡ and is a dense subfield in with respect to . ¥4lGAB;GLF¦Q§ORQSGAT= Every discrete valuation field has a completion which is unique up to an iso- morphism over . ¡ Proof. We verify that the field ¨/ with the valuation is a completion of . is ( embedded in / by the formula ( ) mod . For a fundamental sequence ( ) 0 \ ¡ § Y"\\ © and real , let 0 0 be such that ( ) for all 0. Hence, for 0 ( ) ¡ © ¨/ we have ¡ (( 0 ) ( ) 0) , which shows that is dense in . Let (( ) ) ccc ¡ \ \ be a fundamental sequence in ¨/ with respect to . Let (0), (1), be an increasing ( ) ( ) ( ) Y \ \ \ Y sequence of integers such that ¡ ( ) for , ( ). Then ( ) is a 2 1 1 2 ( ) ( ) ¡ ¡ ¨/ fundamental sequence in and it is the limit of (( ) ) with respect to in . Thus, ¡ ¡ we obtain the existence of the completion / , . ¡ ¡ ¡ ¡ ¡ ¡ ~©( If there are two completions 1, 1 and 2, 2, then we put ( ) = for and x ¡ ¡ extend this homomorphism by continuity from , as a dense subfield in 1, to 1. It is easy ¡ ¡ ¡ ¡ ¡ ¡ ¡ to verify that the extension : 1 2 is an isomorphism and 2 ª = 1. x x ¡ ¡ ¡ We shall denote the completion of the field with respect to by p& or simply . r ¡ 1.7. Examples of complete valuation fields. 1. The completion of with respect to ¢ of r (1.3) is denoted by ¢ and is called the field of -adic numbers. Certainly, the completion of r «1¬S«{ ­ r r with respect to the absolute value of (1.1) is . Embeddings of in ¢ for all prime r and in ­ is a tool to solve various problems over . An example is the Minkowski–Hasse ¥°²±d©)r r Theorem: an equation ®¯¥°²±t³°´td± = 0 for has a nontrivial solution in if and only r ¢ if it admits a nontrivial solution in ­ and in for all prime . Xµ¢ The ring of integers of r-¢ is denoted by and is called the ring of -adic integers. The ¶¢ residue field of r ¢ is the finite field consisting of elements. s t ¡ s t 2. The completion of ( ) with respect to u is the formal power series field (( )) + { ® t ©2s \ of all formal series { with and = 0 for almost all negative . The + { ® ¡ s t t ring of integers with respect to u is [[ ]], that is, the set of all formal series 0 , ©+s s . Its residue field may be identified with . ¡ 3 % & (1.8). Representatives. For simplicity, we will often omit the index in notations & , , * | & & , . We fix a prime element of . ·¸¹% ©· A set · is said to be a set of representatives for a valuation field if , 0 %©º%/1* and · is mapped bijectively on under the canonical map = . Denote by + { ¼ ¼ rep: ©»· the inverse bijective map. For a set denote by ( ) the set of all sequences { + { + ¥;° ° ¥°4©+¼ ¼ ¼ \)½5¤ ( ) , . Let ( ) { denote the union of increasing sets ( ) where . L The additive group has a natural filtration ° ° +1 ccc c ¬¬¬m¾!| %¾(| %¾ ° ° +1 %µ/1| %¿ The factor filtration of this filtration is easy to calculate: | . QiFkjRlN8POm8onp<ACEMm<DORQiGLT+qQS8DCEVLF 4 e0fhg 798;:-:-<= ¡ | © Let be a complete field with respect to a discrete valuation . Let ° for each À À ©+X ¡ | be an element of with ( ° ) = . Then the map + { Ä + { ¥° °´Á0Âà ¥;° |A° Rep: ( ) { ( ) rep( ) { À °´Á0Â$ °ÅÁ0 ¡ ¥° ¥°µ$ is a bijection. Moreover, if ( ¥° ) = (0) then (Rep( )) = min : = 0 . À ¥° Proof. The map Rep is well defined, because for almost all 0 we get rep( ) = 0 and the ® |A° ¥° °´Á0Â$ Æk° °ÅÁ0 series rep( ¥° ) converges in . If ( ) = ( ) and À ©WX ¥°µ$ Æk°D \ = min : = À ¡ ¥ | Æ | \ ¡ ¥ | ]Æ | ,Ç\ ,!\ ° ° ° then ( ) = . Since ( ° ) for , we deduce that c ¥° Æk° \ ¡ (Rep( ) Rep( )) = Therefore Rep is injective. À ¥° ¥°³$ È©§ |9; In particular, ¡ (Rep( )) = min : = 0 . Further, let . Then = with +©Ç3 | 0 0 '©!3 ¥ 0 \©2X , . We also get = for some . Let be the image of in ; +1 $ 6 ¥ | ©6| % then ¥ = 0 and 1 = rep( ) . Continuing in this way for 1, we obtain a ® ¥ | ° convergent series = rep( ° ) . Therefore, Rep is surjective. É GAlGLCEC<LlEÊp= |9 ]© We often take | = . Therefore, by the preceding Lemma, every element can be uniquely expanded as + { Ä À ° °Å| °4©+· ° = and = 0 for almost all 0 {Ë Ë Ë 8;I1QSTmQiORQiGLT9= )6]©Ì|9L% 6Í |9 g If , we write mod . 3 (1.9). Units. The group 1 + |p% is called the group of principal units 1 and its elements are À ° 3 | % called principal units.