On ｌ-extension with given ramification points over a real quadratic field

Title On ｌ-extension with given ramification points over a real quadratic field

Author(s) AMANO, Kazuo

Citation [岐阜大学教養部研究報告] vol.[5]  p.[25]-[27]

Issue Date 1969

Rights

Version 岐阜大学教養部 (Dep. of Math., Fac. of Gen. Educ., Gifu univ.)

URL http://hdl.handle.net/20.500.12099/47407

※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。

On /-ex tension w ith given ram ification points over a real qUadratic field, 犬 十

25

By K azuo A MANO

( D砂. of Ma哨 ., Fac. of Ga . Educ., Gi和 皿 佃.) R eceived Oct. , 31. 1969

B asic consideration つ

ぺV e shaU introducむ the follow ing notation, ゐ : 4 finite algebraic num ber field ゐ(゛ ) : the maximal /-eχtension over ル

(M : the Galois group of the l-extension A砂 )/ ん S : a finite set of finite prim e divisors of ん

ん (; ) : themaximal /-extensionunramifiedoutsideof Sover ゐ ■

(Ms : she Galois gr6up of the /-eχtension カ(; )/ ん.

lt is obvious that the group @ s is the factor group of 6 m odulo the norm al subgroup geherated by the inertia groups of 雫ʼs, w here 串 is a prime divisor in ん(回 and 雫

divides 雫圧 S. /

T herefore (55S ig obtained from the knowledzes of 6 and the inertia groups. M ore precisely, as is w ell know n, the following is a eχact 叩 quence,

1→ U/かFU→ I/がF→ C( カ) /C( ルy→ 1,

w here l is the ideles group, U the unit ideles group, C ( ル) the ideal classes group, F the /-th. pow er of l ,and が the m ultiplicative group (jf 力.

T hen generators of 6 are obtained from the norm residue sym bols for generators

of l/がF. Hence we nlay choose for generators of l/かF the preimages of a basis C )f

l lntroduction

L et / ( ≒2) be a fixed rational prim e num ber and 6 the Galois group of a field

extension 尺/ん. W e shall be interested in these groups in the case where カ is a local

or global field and 尺 its maχimal /, eχtension, i.e., the maχimal normal eχtension w hose Galois group is a pro-/-group. ln the algebraic num ber theory, it is an im por- tant problem to determ ine concretely the structure of these groups. T h沁 problem is cgm pletely solved if ん is a local num ber field CO . But, for an algebraic num berj ield

v v

カ, it is not complete. ln this case, I .R .Safarevic 〔 7〕 has pointed out the interest and im portance of the group (Ms, the Galois group of the m aχim al /-eχtension of カ unramified outside of a set of prim e divisors of 力. ln the present note, w e shan give the structure of the group (Ms for a real quadratic field ゐ and for a special set S.

26 K azuo A mano

the finite cok em el and for each divisor p the basis of Up/ U昿

E speciany, if ル has class num ber one, w e may only consider the generators of

Up/Uが. from now on, we shall only consider the field んwith class number one. For

any idele 包E I , the norrn residue symbol ( 截, ん(1)/ ん) induces the map of U/ が FU2

1/ 炉I ʼ onto (M/砂 〔@ , (S) , where C(M, [M]l is the commutators group of 6 and ん(1) the

maxim al abelian /-extension over ル w ith Galois group 6 / 〔(M, (S) .

By virtue of local field theory, for any p-adic unit 仰 , ( 仰 , が )/ ゐ) is an elem ent of the inertia grop in ゐ(り/ ん. H ence it is unit 如 (Ss for p(三S.

A s above m ention, the studies of 6 s are reduced to study the structure of 即 / Up for p巳 S.

Structure of Gs

L et ル be a real quadratic field with class num ber one over rational field Q, i,e. ,

-

ん= Q( , /j ) for j > o, り the unique fundamental unit in ゐ, and Up the units group in

p-adic fieldʼ郎 for each p. ln addition, we suppose that the set S consists of two

1ヒ )rimèdiVis6rs 夕 and y which are also prime divisors in ん. By virtue of local field theory 〔2, 5〕 , the following lemma is wen known.

LM EMA 1. T he base of Ur/ Uj is { G 2. 1} , w here Cr2 1 1s the prim itive ( が - 1) - th。

-

toot of unit. T he base of U ,/ U 八 s { 1 十 / , 1 十 / ぐ 引 。

-

L EMMA 2. P141 1?ヨ ( 1十 /) “1 ( 1十 1、/ d ) “2 加 U・/ W , z岫 g に 1j ㎡ α2 αa /-αdici㎡ egeγs.

7み凹 zx ・g 加 9 α2 ≒ o ( mod l) .

Pro(げ. Let J be the discriminant of 力. W e put η= £ 土昔と菟 and ηʼ2¯1= サ土グダ涯 , where α, &, / 1, and 召 are rational integers By Fermat theorem, wehavJ ヨ 1( mod /,)

≒1( ゜ od /2) ・ 311d{ F5 0 ( ゜ od /) 。 0 11 the othelʼ h肘1d・ sillce /2¯ 1 1s e゛e11・2゛ e h゛ e ノ12 一丿 召2= 4. Hence 召≒o ( mod /2) . ln other words, α2/≒o ( mod /2) and therfore α2≒o

(mod /2) ,

W e shan denote by ( α) 。 the idele of 力 such that its ♪-compgnent is α and the other 1. W e put

( 1)

F. = ( ( G2-1) 。, 恥/ゐ)

( 1)

石,1 = ( ( 1十/) 。 恥/力)

( 1)

石,2 = ( ( 1十yy7 ) x, 恥/ゐ)

( 1)

where hZk XS the maχimal abelian /-eχtension unramified outside of S over ん. W e

denote by r the preim age of r for the map of 6 s onto (Ss/ 〔(55s,(呂J @ j .

PRoPosI TloN 3、 L et k , S αμj 6 s & αs 功 θ阿 。 7 力四 X加 g 夕・叱 ♪ 6 s fs g 印 a ・ぶ ㎡ 勿

用加ff用α/ り ・ぶ な附 {r。, r。1} .

Proof . lt is obvious that F。, 乱。1 and 乱。2 are generators system. 0 n the other hand,

we have

( 1) ( 1) ( 1) ( 1)

1= ( ( η), 島μ ) = ( 「ζ2」 ) 。, んs/ん)“( ( 1十/) 。 恥/ル)゛ ( ( 1十匈 ¯J ) 。 ゐs/ゐ)“2

¯ ぞTpα Fz4α1 ぞTI・2α12

Since α2≒o ( mod /) , Fj・ and 石・1 are minimal generators system. Therefore, we have

2

On /-eχtension w ith given ramification points over a real quadratic field. 27

0 u r p r op osl tl on .

T he following is w ell know n( 41 .

PROPOSI TI ON 4. L d la be a j 戒 le χ)-d i c e灯 a si回 of QI) n d N ( p) ≡ 7 ( mod l ) . Tha 仙 e G ㎡ ois gro副) Q?) oヂ 独 e ma ;ima川 ・a;teれsion ouer 晦 is α 加 o-/-gγOt41) 面 臨 t加o geneγd oγs (7, て, α11d α s緬 g le γeld ion (7¯̀ で f7= f r 1 肋 hete f7 1s a F r obenivs-

auI omor油 ism o∫ 油e maximal t4回 amij d l-e財回 sion n d 7 a ga erd or oヂ括 e cydic

緬e哨 a 訂 o副) oj (55. ∧

( 1)

W e put i 。 = ( ( カ) 。, 恥 / ん) . T hen it is obvious that ら is a F robenius-autom rphism of the m aximal /-extension over 郎 ・ and ら a generator of the cyclic inertia group of (M.

L EMMA 5 PM I) E ( 1十 /) リ フx U, / UJ, 扨 heγe b is I) -adic i ㎡ egeγ. T hetl uXe halXe b≒ a

( mod l) げ 醐 d O㎡y 仔 j)E I ( mod l) a筧d j) ≒7 ( mod μ) .

Pro昿 Put 夕= Σ の が in (と Then we have that ,70≡1 ( mod /) and α1≒o ( mod が)

are equivalent to 夕三1 ( mod /) and ヵ≒ 1 ( mod /2) . T herefore we have our lemma.‥

T HEOREM 6. £ 政 ゐ, S 皿 d 邨 s be as aboue 皿 d ld j) ≒ 1 ( mod F ) . T hm (脳 isいa j)γ0-1-gγoul) 扨i臨 t肋o gem rd oγs (7. , 71, aれd a s緬 gle rd alio筧 y勁 71μ。= 7・j 2- 1 7=り・OOj T. Since

( 1) ( 1) ( 1) ( t) ( 1)

1= ( ( ♪) 恥/ル) = ( (φ) 1 ), h yk) ( ( ♪) 。 恥μ ) ( 0 ) r, 恥/ル) ( ( 1十/) 。 恥/ん) ゛

- -

¯ j yj r J14 &| ʼ

and lem m a 5 , ?-, 。1 1s generated by ftp , B y virtue of propositi(jn 3 and 4 , w e have 6ur

theorem . ジ

R eferences ‥

1) E.Artin and J.Tate, Class field theory, Hahvard ( 1961) 2) H.Hasse, Zahlentheorie, Berlin Akademie-Verlage ( 1949)

3) Y.Kawada, 0 n the structure of the Galois group of some i:nfinite eχtensions, J. Fac. Sci.。

U niv. of T okyo, Sed . I ,7 ( 1954) 1- 18

4) H.Koch, /-Erweiterungen mit vorgegebenen Verzeigungsstellen, J. f. reine u. angew. Math。

219 ( 1965) 30-61

5) J.-P. Serre, Corps locaux, Hermann Paris ( 1962) 6) J.-P.Serre, Cohomologie Galoisienne, Berlin ( 1964)

v v

7) I .R. Safarevic, Algebraicnumber fields ( Russian,) Proc. lnt. Congr. Math. Stockholm ( 1962) 163- 173

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