C H A P T E R PI^ELIMINAnES In this chapter we collect, for the convenience of the reader, all the known concepts and results i^loh are required in the subsequent chapters. These are divided into sections bearing titles of the topics to which they relate. 1.1. Fitlds with valuation Let Gi toe €m ordered group, nultiplicatively written. Let, further-f- toe an associative operation on G\ such that the distributive laws (1.1.1) ac +• ^c z^(gu+&)<L, (1.1.2) -Ca/4fr<^^<= CCOu^t)^ hold for all CL, -6- , C , in Gi . We shall adjoin a zero element to <?, i.e. we form QU{o] such that (1.1.3) OJ-O- OCL= 0 for all OU € Gi^ (1.1.4) 0 < (h for all a, & G. - 2 - DEFINITION 1.1.1 (BACHMAN [3], p. 72, Definition 2.2). A valuation of a field K is a map 1 \". K-^ 6i U{oJ, \^ei» G iB an ordered group provided (1.1.5) for a G K , (Cbl = 0 if and only L-f CL = Oj (1.1.6) for CO, ^€ K . 10.^1 = la.1 l-6-( ; (1.1.7) for a.-^GK, lCL+-tl^ l^l+l^l. It is always possible to introduce an additive operation in an ordezed group Gi by defining (1.1.8) OJ-^ (^ = 'Yv^doc. CCb , ^) for a/,^G6t. It is easily seen that this -p is associative and that the distributive laws are satisfied, Further, condition (1.1.7) of Definition 1.1.1 becomes (1.1.9) \ch~{-J^\ <, TVLC^ac Clcol , \^0. The valuation in this case will be called non-archlmediaB. Ve shall assume throughout that <j\ ^ C&) • DEFINITION 1.1.2 (BACHMAN SO p.77, Definitions 3.1t 3.2). A subgroup H of <?i , where G] is an ordered group is called isolated if ^ eH, and ^ < 00 ^ (L^ where CL e G\ , imply CU€|-|. The order type of the set of all isolated subgroups of (jj distinct from G\ is called the rank of 6i . DEFINITION 1.1.3 (BACHMAN [3l p. 77, Definition 3.3) If I I is a non-archimedian valuation of a field K into ^^U'foj where Gr is an ordered group, then the rank of \ I la the rank of Gj - 3 - PHOPOSITION 1.1.1 (BACHMAN [3] p. 81, Th«oi»m 3.4). An oTiered group G\ of rank 1 is order Isomorphic to a subgroup of the additive group Tl of real numbers. RBMAHK 1.1.1. In riew of Proposition 1.1.1, in consider­ ing valuations of rank 1, we may assume that the ordered group Gt is a subgroup of the additive group of real numbers with the usual ordering. We shall be concerned with valuations of rank 1 only. A valuation of rank 1 may thus be defined as follows. DEFINITIOH 1,1,4 (BACHMAH [3] p.5, Definition 2,1). A valuation of rank 1 of a field K is a mapping | I : from K into the reals 1\ such that for all CL , ^ e k ^ (1.1.10) \(Xj\7'yO and \ou\-0 if and only if CO = 0 ; (1.1.11) 1 OuL[ - fCLl 1^1 •. (1.1.12) I CL-f ^1 ^ la.l+1^1.; We shall generally omit, for the sake of brevity, mention­ ing that the valuation is of rank 1 explicitly and simply speak of such a function as a valuation, it being understood that such a map is always into the reals'Bv. K is called a field with a valuation or a valued field. If (1.1.12) is satisfied in the stronger foim (1.1.13) I Cb-+- Qy\ < -ma^c c \ou\ ,1^0, a., ^ e K^ then it is called a non-archimedian valuation. - 4 - We give some examples of non-aichlmedlan valuations. EXAMPLE 1.1.1. Por any field K , we define \0\~0 and \aj\=-\ toT all 0/4^0 in K . This is called the trivial valuation on k . Valuations other than the trivial valuation are called non-trivial valuations. EXAKPLS 1.1.2, Let Q denote the field of rational numbers, c a fixed real number such that 0-cc< I and p a fixed prime number. If x is any rational number other than 0 we can write where CL , -^ , oC are integers, [^ "fa, ^ j't-S- • We define (1.1.14) l^lb= C°^ O/^ct 10((,= 0. It is known (BACHMAH [3l , p.l) that ( (. is a non- archimedian valuation on Q and is called the \:> -adic valuation of Q. EXAMPLE 1,1.3. Let K be any field and KLt]be the ring of polynomials over K . If f>Cfc;i8 an irreducible polynomial in K Lb] then we define (1.1.15) locctM.= , ? where (XCb), -^Cb)Gl<Cb3, \=>'\ CL . f^t"^* ®"^ C is a m» ^ mm fixed real niunber In ( 0 » X )• (1.1»15) gives a non- archimedian valuation on KUb] and is called a ^Cb)- ~adio valuation of KLtUCsee Bachman [5] p.14). It follovnB immediately fiom Definition 1.1.1 that a valuation satisfies the following conditions (1.1.16)1 11 = I • ; (1.1.17)1-11 =^i ^ (1.1.18) 10.^1= I 0.11^1-' ; (1.1.19) |i<5^l-l^i|^ \aj-U. PPOPOSITION 1.1.2 (BACHMAH [?] pp.7» 8). If | | is a non>archiffledian valuation and if Icol > \£>,\ then \(Xy+4>-\~\(l>\ More generally, we hare . COH)ItLABY 1.1.1. For a non-archimedian valuation we have and I a,+ a2_+--- + clw| = I Ob, I -il^ 10.^1^ 10/(1 -for j.= i-.5,-.Ti The Corollary is a consequence of finite induction an Yl using Proposition 1.1.2. * We denote by 1 the unit element of K as also the real number 1, with, of course, no room for confusion. - 6 . The following characterization of a non-aiohlffledlan valuation will be freely used in the sequel. PROPOSITION 1.1.3 (HAMANATHAH [23] , p.204). A Taluation | | on a field K is non-arohimedian if and only If (1.1.20) 1 Til ^ 1 for eveiy Integer yi e K * ThlB shows immediately that COK)LLA!?y 1.1,2 (RAMANATHAN §3] , p. 204). All the valuations of a field of characteristic j=>=|=0 are non- archimedian. We continue to assume that I i is a non-aichimedian valuation and we consider the set Sy of all cu€ K such that I CU| ^ 1 • ^®* ^ denote the set of all cue K such that lal< 1- PHOPOSITIOH 1.1.4 (BACHMAN [3] , p. 8. Theorem 2.3). If I ( is a non-archimedian valuation, then the set V is a zing with unit element and 'P is the luiique maxiaal ideal of V and hence is a prime ideal. DEPIHITIOH 1.1.5. The ring V (defined above) is called the valuation ring associated with the non-archimedian valuation | j • The field \^ is called the associated residue class field. * Throughout we write \r\\ v^ere n is an integer, for Ini/ (1 being the unit element of K ). - 7 - (1.1.11) of Definition 1.1.4 showe that CL -^ (Ctl 1» a homomorphism of the multiplicative group K of non-sseio elements of K into the multiplicative group of positive reals. This homomorphic image is a group | K ( which we call the value group of K (see MMA.HATHAN [23l p,213), PEFINITION 1.1.6. The valuation | j is called discrete or dense according as the value group of K is discrete or dense. HEMAPK 1.1.2. A valuation is either discrete or dense, A valued field K can be viewed as a metrLo space with the distance function P defined by (1.1.21) pccb,t)= |a-^l satisfying (1.1.22) pCa..-6L);^0 and PCa,6.) = 0 if ana only If (X - ^ ', (1.1.23) pca,^)= pc-^.o,): (1.1.24) Pca,^) ^ PCa.,c)-i-pCc,l), Ca,4,c£K) If, in addition* K is a non-arohlmedian valued field then this distance function P can be easily seen to satisfy the following ultrametrlc Inequality (1.1,25), the stronger form of (1.1.24) (1.1.25) ?COL,t) ^ 7na:>c (pCCL, c) , pcc, 6.)) • Thus a valued field K is a metric space. - 8 - THEOREM 1.1.1 (BHJHAT [>] , p.18). Let K be a field with a non-trivial non-archimedian raluation. Then the following conditions axe equivalent. (1.1.26) K is locally-coapact. (1.1.27) The valuation ring V of K is compact, (1.1.28) K is ^ complete metric space. ( | is a discrete valuation and the residue class field is a finite field. The above metric p gives rise to the notion of conver­ gence in K as follows. If {jJir\,] is a sequence of elements of K then it is said to converge to the element CU in K » i^f for every real number £. > 0 , there exists an integer Nce) such that la^-Cul < ^ for -rv y^ NCe) . In terms of the metric, more explicitly, this means that PCdyu, Cl/)-> 0 , •Xi _^ CO • The sequence •[Q./>^] is said to be a Cauchy sequence if for every real number 6. > 0 there exists an UC&) such that \Ci'y[-(X'Yv\\ <S whenever ru.YW ;:> NCE-j. In tenas of the metric this means PCd^, OL'TA) -> 0 as n,'rv\ -> oo , independently of each other. However, it is clear from the following proposition that the independence of VL , rrt in the above definition is of no significance in a non- archimedian valued field.