Cross-Sections for p-adically Closed Fields

Transcription

1 JOURNAL OF ALGEBRA 183, ARTICLE NO Cross-Sctios for -Adically Closd Filds Phili Scowcroft* Wslya Uirsity, iddltow, Cocticut Couicatd by Loard Lishitz Rcivd Dcbr 1994 INTRODUCTION If F is a fild with o-archida valuatio : F G, a cross-sc- tio for th valud fild Ž F,. is a grou hooorhis : G F that is a right ivrs to. Soif is ri ad : Q Z is th -adic valuatio, is a cross-sctio for Ž Q,.. Aothr class of xals ariss fro filds of gralizd owr sris 9,. 23 : if F is ay fild, Ž,. is ay ŽŽ.. ordrd Ablia grou, ad F t is th fild of owr sris Ýat with cofficits a F ad suort : a 04 wll-ordrd by, th FŽŽt.. has valuatio Ý a t 0th last with a 0 ad cross-sctio t. If is ri, a -valud fild 8,. 7 is a valud fild, of charactristic zro, i which has iial ositiv valu ad whos rsidu fild has lts. A -valud fild Ž F,. is -adically closd just i cas o algbraic xtsio F of F with valuatio xtdig is -valud. Th class of -adically closd filds is to Ž Q,. as th class of ral-closd filds is to R: s 8, Sctio 1 for a discussio of aalogis btw ths classs of filds. I thir fudatal work o -adically closd filds 1, Ax ad och foud it covit to work with -adically closd filds havig * Th rsarch rortd hr was suortd by NSF Grat DS $18.00 Coyright 1996 by Acadic Prss, Ic. All rights of rroductio i ay for rsrvd.

2 914 PHILIP SCOWCROFT cross-sctios. Crtai -adically closd fildsthos that ar sudo-colt 1,. 614always hav cross-sctios 1,. 636, ad i fact oralizd os: cross-sctios that sd th grou lt of last ositiv valu to. Chrli 4, latr showd that all 1-saturatd -adically closd filds hav oralizd cross-sctios, but o furthr study of cross-sctios was dd to dvlo th gral thory of -adically closd filds. Wh studyig sialgbraic uivalc rlatios ovr Q, I grw itrstd i fidig a -adically closd fild without a cross-sctio. Discovrig o i th litratur, I vtually built such a fild, ad foud cssary ad sufficit coditios for th xistc of Ž oralizd. cross-sctios. Ths coditios aar, aftr th rliiary las of Sctio 1, i Sctio 2, ad ar followd i Sctio 3 by a roof that ay cross-sctio for Q xtds to a cross-sctio for ay -adically closd xtsio of Q. Sctio 4 th shows that if Ž F,. is a -adically closd fild i which Ž Q,. dos ot bd, th Ž F,. has a -adically closd xtsio without a cross-sctio. Sctio 5 rfis this rsult by roducig, aog othr xals, -adically closd filds that hav cross-sctios but lack oralizd os. Th roofs of ths rsults will us odl-thortic tchius xlaid i 3. I a gratful to A. acityr ad A. Pillay for hlful suggstios at a arly stag of y work. 1. SOE BACGROUND INFORATION Th Itroductio sigls out -adically closd filds as axial lts of a artially ordrd class of -valud filds. O ay also charactriz a -adically closd fild i trs rfrrig oly to th fild itslf 8,. 34. A valud fild Ž F,. is -adically closd just i cas it is -valud ad Hslia ad its valu grou is a Z-grou; a Z-grou is a ordrd Ablia grou G, with last ositiv lt 1, such that for ach ositiv itgr GG has xactly lts. So triology fro 8, 2.2 will rov usful i what follows. Lt Ž F,. b -adically closd. If G is F s valu grou ad 1 is th last ositiv lt of G, o ay idtify th subgrou gratd by 1 with Z. It is a covx subgrou of G, whos uotit by Z bcos th valu grou of th coars valuatio : F G GZ of Ž F,. Ž is th usual uotit a.. Sic vry ratioal ubr has coars valu zro, Ž F,. s rsidu fild F th cor fild of Fhas

3 CROSS-SECTIONS 915 charactristic zro. If R is th valuatio rig of Ž F,. ad x R x F is th rsidu a, o ay dfi a valuatio : ŽF. Z of F by th coditio Ž x. Ž x.. Exloitig aalogous rortis of Ž F,., o ay show that ŽF,. is -valud ad Hslia: so ŽF,. is -adically closd. Bcaus Q Q ad th rsidu fild of F is th fild with lts, ay lt of F is th -adic liit of a suc of ratioal ubrs: so o ay bd ŽF,. i Ž Q,.. Bcaus ay bddig of ŽF,. i Ž Q,. is th idtity o Q, ŽF,. is isoorhic to a uiu valud subfild of Ž Q,.. O ay thrfor viw ŽF,. as a valud subfild of Ž Q,.. Yt sic Ž F,. is Hslia of rsidu charactristic zro, o ay xloit Zor s la to fid a fild F R isoorhic to F udr xx 7, So as valud filds, Ž F,. ad Ž F,. ar isoorhic. Ž F,. ay cotai ay valud subfilds Ž F,. with this rorty, but sic all ar isoorhic to a uiu valud subfild of Q, arbitrary slctio of a sigl such Ž F,. allows o to viw Ž F,. as a valud subfild both of Ž F,. ad of Ž Q,.. Ž Fially, if F, ad F,. ar -adically closd filds ad g : F F is a bddig of filds, th g icluds a bddig g : G G of valu grous, rlativ to which g is a bddig of valud filds. This rsult holds for two rasos: if Ž F,. is -adically closd ad x, y F, 2 2 x y just i cas x y is a suar i F Žwh 2; wh , x y should b a cub i F. 2,. 4 ; ad th thory of -adically closd filds i th laguag of rigs is odl colt 8, O ay thrfor rstrict atttio to fild bddigs, istad of valud-fild bddigs, i th class of -adically closd valud filds. 2. A CRITERION FOR CROSS-SECTIONS Lt F, b -adically closd. For ach 2 lt F 4 P x F : x is a th owr i F 1 Rfrc 8,. 86 rovs th odl-coltss of th thory of -adically closd filds i a laguag that xtds th laguag of rigs by a fuctio sybol for divisio ad a rdicat sybol for th valuatio rig. Sic th forula Ž xy. is uivalt to th forula Ž y 0&Ž 0.. Ž y0&zžž z. &yz x.., o ay liiat all occurrcs of at th cost of itroducig w xisttially uatifid variabls. Also, both th valuatio rig ad its colt ay b dfid by xisttial forulas i th laguag of rigs. Thus th thory of -adically closd filds i th laguag of rigs is also odl-colt.

4 916 PHILIP SCOWCROFT ad P F P F ; 2 th surscrit F will b drod wh thr is o dagr of cofusio. F F Each P is a subgrou of fiit idx i F, 2,. 5 ad so P is a Q subgrou of F,. If x P, th Ž x. Z is divisibl by vry ositiv itgr ad so is zro; also, th iag of x i ay uotit rig Q Z Z ust b a uit ad a Q th owr i Q s grou of uits Q Q ; thus x is 1 odulo for all 1, ad P 1 4. I gral, Ž F LEA 1. P,. is a diisibl subgrou of ŽF,.. Proof. Say x P ad 2. If x is ot divisibl by i P, th o of x s th roots i F blogs to P. If ths th roots ar y 1,..., y k, th for ach l 1,...,k thr is l 2 such that yl P l.so if Ł,o yp. Yt sic x P, x z for so z F l l l, ad z P is o of th th roots of x. This cotradictio ilis that P is divisibl. F Lt R b th valuatio rig of F, ad U U b th grou of uits of R. Ž LEA 2. If U, th. Z for so F. Proof. By th idtificatios ad i Sctio 1, F is isoorhic via xx to th rsidu fild of Ž F,..Soif U,Ž. 0, Ž. 0, ad Ž. for so ŽF.. Thus Ž. 0 ad Ž. Z. Not that F U. If U, say that x F is of -sort just i cas for ach 2, x lis i th sa cost of P as so owr of. Lt F S S x F : x is of -sort 4. l Sic : l Z4 ad th P ar subgrous of ŽF,., S is also a subgrou. Ths dfiitios allow o to stat THEORE 1. Ž F,. has a cross-sctio if ad oly if F SU for so U; i this cas, Ž F,. has a cross-sctio sdig 1 to. Proof. Assu that Ž F,. has a cross-sctio. Sic 1 1, 1 for so U. If xf, th x x Ž x., x

5 CROSS-SECTIONS 917 whr x x U bcaus x x; so th roof will b colt if x S. Wh Z ad, blog to F s valu grou G, writ wh od for so G.. If 2, th sic G is a Z-grou thr is a itgr l 0, such that x l Ž ll1ig.. So thr is y F ad od for which x l y Ž l y., l l l x y Ž. Ž y. Ž. Ž y.. Thus x ad Ž. l blog to th sa cost of P. Sic 2 is arbitrary, x S as dsird. Now assu that F SU for so U. Lt U U F ad U xu:ž x1. Z 4. U ad U ar subgrous of Ž U,., ad U U14 sic U F Q.If xu, th La 2 rovids x U with Ž x x. Z :so ŽŽ xx. 1. Z, xx U, ad x x Ž xx. U U. Thus U is th itral dirct roduct of U ad U. By Hsl s la i Ž F,., U U P. Covrsly, if x U P, La 2 rovids Ž. ŽŽ x U with x x Z; so xx. 1. Z, xxpby Hsl s la, x P sic x P, ad x 1 sic x F Q ad P Q Q P 1 4. Thus U U P, ad La 1 ilis that Ž U,. is divisibl. Zor s la rovids a subgrou T of S axial with rsct to th l followig rorty: Ž. : l Z4 T ad T U 14 Žot that U U P S.. Suos x S TU. Bcaus T is a ror subgrou of th grou ² T x4: S gratd by T x 4, ² T x4: U 1, 4 ad thr ar t T, l Z 0 4, ad u U 14 for which tx l u. l l l Sic U is divisibl, thr is u U with u u, ad so tx u ad 1 l t x u w whr w x u S.If wtu, a subgrou of S, th x w u Ž TU. U TU :so wtu, ad as abov thr ar t T, Z, ad 1 1 l

6 918 PHILIP SCOWCROFT 4 u U 1 for which t w u. l l l l l l Thus t w u ad t t u. Sic t, t T, u U, ad TU1, 4 Ž u. 1. Bcaus Ž F,. Ž F,. ar -adically closd, F is algbraically closd i F, ad u F :so u UF 1 4, cotrary to th choic of u. This cotradictio ilis that S TU is th itral dirct roduct of T ad U. By hyothsis, thrfor, F Ž TU. U Ž TU.Ž UU. TUU l Ž. I fact, F is th itral dirct roduct of T, U, ad U. Suos that tt, u U, u U, ad tu u 1. Bcaus t uu U,t P: for if t P for so 2, th sic t S, t P for so l Z, dos ot divid l i Zothrwis, t P ad t lod, cotrary to th fact that Ž. t 0. Thus t T U P T U 1,uu 4 1 i th dirct roduct U U, 1 2 ad u1 u2 1. So for ach x F thr is a uiu t T with x t. Clarly, th, thr is a a : G F sdig ay x to th uiu t T with Ž x. Ž t.. Sic is a hooorhis ad T has valu 1, o ay asily chck that is a cross-sctio for Ž F,. that sds 1 to. Not that if, U ad Ž. Z, th S S : for sic ŽŽ. 1. Z, Hsl s la ilis that P, ad so Ž. l Ž. l P for vry l Z ad Ž. l, Ž. l always li i th sa cost of ay P. La 2 ad Thor 1 thus cobi to show that if Ž F,. has a cross-sctio, it has a cross-sctio sdig 1 to for so F. Th rstrictio of this cross-sctio to Z givs a cross-sctio for ŽF,.; so o ay study th cross-sctios for Ž F,. by askig which cross-sctios for ŽF,. xtd to cross-sctios for Ž F, ADICALLY CLOSED FIELDS CONTAINING Q This sctio is dvotd to a roof of THEORE 2. If Ž,. is a -adically closd fild i which Q is bddd, th ay cross-sctio for Ž Q,. xtds to a cross-sctio for Ž,.. Ax ad och rov that ay -sudo-colt -adically closd fild has a oralizd cross-sctio 1,. 636, but thy ivok -sudocoltss oly to bd Q i th fild udr cosidratio. So uch of th rst rsult is ilicit i thir work, though th roof hr rocds alog diffrt lis.

7 CROSS-SECTIONS 919 Proof. Rarks at th d of Sctio 1 allow o to rgard Ž Q,. as Q a valud subfild of,. Each P is a o subgrou of Q of fiit idx, ad o ay choos cost rrstativs fro Z 2,. 5. So for ach 2, lt th a b fiitly ay itgrs, with Ž a., i, i. Q 0,, that rrst th distict costs of P i Q. Th odl-co- ltss rsult tiod i Sctio 1 ilis that for ay -adically closd, th a srv as cost rrstativs for P i, i. Ž. Q Lt b ay cross-sctio for Q. 1 for so U.If S U, th Thor 1 ilis that has a cross-sctio sdig 1to, ad this cross-sctio crtaily xtds. So, suos x. For ach 2 lt k b th uiu itgr i 0,. with Ž x. k od ad b a rrst th cost of P dtrid by xž. k, i. Sic divids ŽxŽ. k. i Ž. ad Ž b. 0,., Ž b. 0 for all 2. Clarly x b! P! bp k! Ž. 2 for ach 2, ad so th stc ž Ž./ y 0 Ž y. 0 od!& z b z y Ž 1. 2 is tru i Ž,.. Bcaus Ž Q,. Ž,. ad ThŽ Q,. ThŽ,. is odl-colt, 2 Ž Q,. Ž,. ad ach stc Ž 1. is tru i Ž Q,.. Thus ach stc ž Ž./ s0 Ž s. 0 & z b z s Ž 2. 2 Žy. is tru i Q, : for if y Q satisfis th body of 1, s y satisfis th body of Ž. 2. So for ach 2, U Q b P Q. 2 Bcaus Ž Q. Q Q U, is a coact grou ad ach U b P is a clo subst of U Q, U Q bp Q. 2 2 odl-coltss of th thory of -adically closd filds i a two-sortd laguag aroriat to Ž,. follows fro th odl-coltss rsults alrady uotd, sic both s valu grou ad ar itrrtabl i th rig.

8 920 PHILIP SCOWCROFT If u is a lt of this st, th sic Q, u U ad k x u P for ach 2. So for ach 2, xu is i th sa cost of P k, xu S, ad x S U as dsird. as 4. FIELDS WITHOUT CROSS-SECTIONS Th xt two sctios will show that Thor 2 dos ot xtd to -adically closd filds i which Q dos ot bd. Startig with a -adically closd Ž F,., both sctios will fid a -adically closd Ž, w. that cotais both Ž F,. ad a scial a ifiitsial with rsct to F : i.., waw Ž x. Ž x. for all x F. Atttio will th shift to L, th sallst -adically closd subfild of that cotais FŽ a.. Bcaus th thory of -adically closd filds i th laguag of rigs adits dfiabl Skol fuctios 5,. 627, L will b th dfiabl closur hr, th algbraic closurof FŽ a. i. A scial dscritio of L s lts will rov usful i what follows. For ach 2 thr is a forula i th laguag of rigs that dfis, ovr ay -adically closd fild, a cotiuous th root fuctio o th st of th owrs; i what follows ' x will b th valu of this fuctio at th th owr x. With th hl of this otatio o ay stat LEA 3. Lt Ž F,. Ž, w. b -adically closd filds. If a is ifiitsial with rsct to F ad L is th algbraic closur of FŽ a. i, th Ž L, w L. is -adically closd, ad for ry g L thr ar c F, u L, ad itgrs 2 ad for which ž ', i / g c aa u, whr a a P ad wž u 1. ŽF., i. Th rsult follows idiatly fro Thor 1 of 10 ad wll-kow rsults i th odl thory of -adically closd filds; th roof will ot b giv hr. Now for th roof of THEORE 3. If Ž F,. is a -adically closd fild i which Q dos ot bd, th Ž F,. has a -adically closd xtsio without a cross-sctio. Proof. Assu that Q dos ot bd i th -adically closd fild F. Sic F s cor fild F F is a subfild of Q, F ust b a ror

9 CROSS-SECTIONS 921 subfild of Q. Q is th oly subfild of Q that cotais vry -adic ubr of valu zro: so thr is a Q F of valu zro. For ach 2 lt b a rrst th cost of P Q, i to which a blogs, ad lt 4 4 z 0 b z y : 2 y b : bf, a st of forulas i th laguag of Ž F,. F. Ay fiit subst of ay b satisfid i Ž F,. F : for if 2 is arbitrary ad b F has ositiv valu, th ad b b! b P F! 2 Ž!. b b! Ž b!. Ž b.; Q ot that vry b 0 sic a b P, Ž a. 0, ad Ž b. 0, k. k k k k. Th coactss thor thrfor rovids a -adically closd xtsio Ž,w. of Ž F,. i which so b satisfis vry forula of. La 3, ad th odl-coltss of ThŽ F,. ThŽ, w., allow o to assu that is th algbraic closur of FŽ b. i ; so La 3 also dscribs all th lts of. Bcaus b b P ad ach wb 0, wb is divisibl by vry ositiv itgr. So if, w has a cross-sctio, wb P bcaus is a hooorhis. O ay thrfor show that Ž, w. lacks a cross-sctio by showig that o g with th sa valu as b blogs to P. Assu, to th cotrary, that g P has th sa valu as b. La 3 ilis that thr ar c F, u, ad itgrs 2 ad for which wu1w ŽF. ad So ž ' / g c bb u. ž ž / / wž b. wž g. w c' bb u wž c. wž b.. Ž Bcaus wbw F. ad c F, 1 ad g cbb 1 u.

10 922 PHILIP SCOWCROFT Sic wu1 is gratr tha ay itgr, Hsl s la lacs u i 1 1 P, a subgrou of, :so cbb P ad b ad c b blog to th sa cost of vry P. Bcaus wž g. wb,wc Ž 1 b. 0. La 2 thus rovids d F with wc Ž 1 bd. Z. Hsl s la agai i- lis that c 1 b ad d blog to th sa cost of vry P, ad so b ad d blog to th sa cost of vry P. Sic b satisfis, d bp, ad so d 2 b P F bcaus F is -adically closd. Thus 2 d b P Q. Q 4 Q Sic P 1 ad a bp, adf, cotrary to th choic of a. This cotradictio ilis that o g P has th sa valu as b ad colts th roof of Thor FIELDS WITHOUT NORALIZED CROSS-SECTIONS Ž If F, is -adically closd, so is F, F. Ž F,.. Sic F Q, th cross-sctios for ŽF, F. ar i o-to-o corrsodc Ž. 1 with F s lts of valu o. Wh F Q, Sctio 3 shows that vry cross-sctio for ŽF, F. xtds to a cross-sctio for Ž F,.; wh F Q, Sctio 4 shows that o cross-sctio for ŽF, F. d xtd to a cross-sctio for Ž F,.. By xloitig or dtaild iforatio about Q s ultilicativ grou, th rst sctio will show that othr cass ay also occur: for xal, thr ar -adically closd filds, with cross-sctios, that lack oralizd cross-sctios. I th rig Z of -adic itgrs lt b a riitiv Ž 1. th root of uity Ž or 1, if 2,Gb. th cyclic subgrou of ŽQ,. gratd by, ad 1 if 2 ½ 1 2 if 2. Accordig to 6,. 246, ŽQ,. is isoorhic to th dirct roduct of Ž Z,., G, ad Ž Z,.; th aig Ž,, a. a

11 CROSS-SECTIONS 923 is a isoorhis of Z G Z oto Q. Bcaus th oly lts of ZGZ divisibl by for all 1 ar th lts of 04 G 04 Ž 0, 1, 0.4, if 2 o ay coclud that Q LEA 4. P G Žor 1,if a 4 Q Q a Not also that sic N is ds i Z, : Z, G, ad a N is ds i Q. Bcaus th costs of ach P ar o i Q, o ay choos rrstativs for th costs of P fro : 0,., G, ad a N 4. Sic th lts of this st ar algbraic -adic ubrs, thy blog to vry -adically closd fild ad ca srv as rrstativs for th costs of P F i ay -adically closd F. Th roof of La 3, which tios cost rrstativs a, i of P, works for ay rrstativs that ar algbraic -adic ubrs with valus i. a 0, ; so o ay assu that th a s blog to : 0,., i, G, ad a N 4. O ay ow stat THEORE 4. Lt Ž F,. b a -adically closd fild, i which Q dos ot bd, which has both a oralizd cross-sctio ad a cross-sctio sdig 1 to. Thr ar -adically closd xtsios Ž,. ad Ž L,. of Ž F,. L such that Ž,. has a cross-sctio sdig 1 to, but o oralizd cross-sctio, whil Ž L,. L has a oralizd cross-sctio, but o cross-sctio sdig 1 to. Proof. Bcaus siilar arguts roduc Ž,. ad Ž L,. L, oly Ž,. will b studid i dtail. Lt Ž F,. b as dscribd. As i th roof of Thor 3, o kows that F is a ror subfild of Q. Lt ½ 1 Z if 2 S 1 2 Z if 2. Sic Q is th sallst subfild of Q that cotais S, thr is a lt asf. By 6,. 246 thr is a suc k 4 1 of atural ubrs for which a li k Q 4 Q i Q. Sic P 1 ad ach P is a o subst of Q, o ay suos that a k P Q for ach 1. Ž 1.. If l 0, is cogrut to k odulo for ach 1, th Ž. 1 also holds with k rlacd by l.

12 924 PHILIP SCOWCROFT Lt k od : 14 Ž f. : f F ad Ž f. 04, 1 a st of forulas, i th laguag of ordrd Ablia grous, with aratrs fro ŽF.. Lt b a cross-sctio for Ž F,. with Ž 1.. O ay show that is cosistt with ThŽ F,,. 1 F by showig that vry fiit subst 4 4 k od,...,k od c f 1 1 c k is satisfid i F,, F. Each a is a th owr, ad if 1 :so k ck d is a d th owr whvr 1 d c. Alyig th ivrs of th isoorhis dscribd at th start of this sctio, o cocluds that d divids k k i Z wh 1 d c. Sic d is a c d owr of ad kc k d is a itgr, d divids kc k d i Z wh c Ž 1dc. Thus k f F. satisfis : wh 1 d c, k c 1 c c d c c c d c f k k od, ad sic f 0 ad k 0, k Ž f. c Ž f. Ž f.. Th coltss thor rovids a -adically closd fild Ž,,. Ž F,,. ad a lt Ž. that satisfis 1.If b, th Ž b. Ž f. for all f F ad k b P for all 1 Ž bcaus :. is a hooorhis sdig 1 to. For ach 2 lt b a, i rrst th cost of P to which b blogs. If k 2, b for so G ad itgrs 0,. ad k, ad thr is w for which bw b.so b w b k Ž. Ž.Ž. b Ž. sic, Z ar uits ad is a cross-sctio for Ž,. that sds 1 to. O ay thrfor writ b b k k Ž w. k b ' k bb, ad thr is a algbraic uit Z for which bb w. '

13 CROSS-SECTIONS 925 As i th roof of Thor 1, w is of -sort: so for ach 2 thr is a itgr d 0,. for which, d, ' bb P. Of cours, b is also of -sort, ad so for ach 2 d b P for so itgr d 0,.; ot that d l for ach 1. O cocluds that b satisfis d ½ Ž. 5 z 0 z y : 2 Ž y. Ž f. : ff ad Ž f. 04 d, z0 Ž. z ' yb :, 2 ½ ž / 5 i. Lt b th algbraic closur of FŽ b. i ad. Ž,. is a -adically closd xtsio of Ž F,., ad by odl-coltss b still satisfis i Ž,. Ž ot that th aratrs i ar ithr algbraic -adic ubrs, ad so i F, or blog to ŽF... To show that Ž,. has a cross-sctio sdig 1 to, o ay ivok Thor 1 ad show that SU.If x, La 3 says that ž ' / x c bb u, whr c F, 2 ad ar itgrs, u, ad Ž u 1. ŽF.. Bcaus b satisfis, bb S. Sic is a algbraic uit, ' ' ' U, bb SU, ad bb usu. By hyothsis, c F S F U F SU:so xsuas dsird. But Ž,. dos ot hav a oralizd cross-sctio. Othrwis, Tho- r 1 ilis that b gu for so g S ad u U, ad by La 3 1 ž ' / g c bb u for so c F, itgrs 2 ad, ad u with Ž u 1. ŽF.. Bcaus u U, Ž b. Ž g. Ž c. Ž b. Ž b. Ž c. Ž Ž b. Ž b..;

14 926 PHILIP SCOWCROFT Ž. so sic b F, 1 ad g cbb 1 u, 1 whr cb u U. By hyothsis, thr ar itgrs s 0,. such that g s P for all 2. d Sic b P for all 2, ach d 1 1 s cbb u b P, 1 d d s. 1 d ad so ach cb u P. Now, d s 0,, ad cb u U:so ds0 for all 2, ad ubg bž. Ž g. P d s d d for all 2. Sic u is a uit, La 3 ilis that u cu for so F Ž. k cu ad u with u 1 F.SouP by Hsl s la, c d P for all 2, ad c d P F sic c, F ad Ž F,. Ž,.. La 2 rovids h F U with Ž ch. Z:so h d P for all 2 by Hsl s la. Lttig, o fids that h l P for all 1 sic l d. Bcaus a, h Q ad a l P Q for all 1, o ay coclud that ah 1 Q P G 1 by La 4. Now, G is cotaid i ay -adically closd fild: so ahg F, cotrary to th choic of a. Thus Ž,. lacks a oralizd cross-sctio. To build th -adically closd xtsio Ž L,. of Ž F,. L that has a oralizd cross-sctio but lacks a cross-sctio sdig 1 to, o

15 CROSS-SECTIONS 927 aks just a fw chags i th rvious argut. So, is ow a k oralizd cross-sctio for F,. Thus b P for all 2 ad Ž w. b i th first coutatios cocrig b. Also, ad d, ' bb P b d P for ach 2; so i th dfiitio of, ach should b rlacd by. If L is th algbraic closur of FŽ b. i ad L L, o argus, as bfor, that Ž L,. L has a cross-sctio, but ow it is oralizd. Yt Ž L,. L has o cross-sctio takig 1 to bcaus thr ar o g S L ad u U L for which b gu. For if such g, u xist, ad s L g P for all 2, th o ay rat th coutatio of bg with ad itrchagd, ad d rlacd by its ivrsto 1 d L coclud that u P for all 2. So if h F ad Žu 1 h. Z, o obtais a cotradictio as bfor. Not that o ay assu that th cross-sctios for Ž,. ad Ž L,. L tiod i Thor 4 xtd th rlvat cross-sctios for Ž F,.. If, L for xal, is a oralizd cross-sctio for F,, lt T S1 b a subgrou axial aog thos that cotai raž. ad itrsct UL trivially, ad argu as i th roof of Thor 1 to obtai a cross-sctio for Ž L,. L whos rag is T; this cross-sctio will b oralizd ad xtd. A QUESTION Is thr a iforativ classificatio of th cross-sctios a -adically closd fild ay hav? A sil attt at such a classificatio associats, with ay -adically closd Ž F,., th st CŽ F. of all a F such that so cross-sctio for Ž F,. sds 1 to a. By Sctio 2, Ž F,. has a cross-sctio if ad oly if CŽ F.. Th rsults of Sctios 3, 4 show that if Q bds i F, th CŽ F. F Z Z, whil if Q dos ot bd i F, th CŽ. for so -adically closd F. Sctio 5 shows that a oty CŽ F. d ot b a subgrou of Z sic 1 d ot blog to CŽ F. ad that th class of all CŽ F. is oly artially ordrd by iclusio. O ca show that a oty CŽ F. is a subgrou of Z just i cas CŽ F. is closd udr ultilicatio, but o ight also wodr whthr CŽ F. is a subgrou just if it cotais 1. Ad ulss thos CŽ F. s that ar ot grous hav so othr itrstig structur, o ay also wat to fid a w objct that bttr dscribs th cross-sctios of Ž F,..

16 928 PHILIP SCOWCROFT REFERENCES 1. J. Ax ad S. och, Diohati robls ovr local filds, I, II, A. J. ath 87 Ž 1965., , L. Blair, Substructurs ad uifor liiatio for -adic filds, A. Pur Al. Logic 39 Ž 1988., C. C. Chag ad H. J. islr, odl Thory, North-Hollad, Astrda, G. Chrli, odl Thortic Algbra: Slctd Toics, Srigr-Vrlag, Brli, L. va d Dris, Algbraic thoris with dfiabl Skol fuctios, J. Sybolic Logic 49 Ž 1984., H. Hass, Nubr Thory, Srigr-Vrlag, Brli, S. och, Th odl thory of local filds, i Lctur Nots i ath. Vol. 499, Srigr-Vrlag, Brli, A. Prstl ad P. Routt, Forally -Adic Filds, Srigr-Vrlag, Brli, O. F. G. Schillig, Th Thory of Valuatios, Arica athatical Socity, Nw York, P. Scowcroft, or o iagiaris i -adic filds, J. Sybolic Logic, to aar.