Title On the p-Class Groups of Relatively Abelian Number Fields
Author(s) OHTA, Kiichiro
Citation [岐阜大学教養部研究報告] vol.[23] p.[21]-[23]
Issue Date 1987
Rights
Version 岐阜大学教養部 (Dept. of Math., Fac. of Gen. Educ., Gifu Univ.)
URL http://hdl.handle.net/20.500.12099/47609
※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 the p-Class Groups of R elatively A belian N umber Fields
21
K iichiro OH T A
1. 1n general, let ん be an algebraic number field of finite degree, and let X, be a rational prime number. T hen, the /) -Sylow subgroup of the absolute ideal d ass group of ヵ w ill be called the y d ass group of ん and will be denoted by G ( ♪) , whose order will be denoted by /4 ( ♪) . M oreover, let 尺 be a GaIois eχtension of finite degree over ル . T hen, the sub政 oup of all ideal d asses of G ( ♪) which are ambigousʻwith resped to K Zk will be called the ambigous p-d ass group of K with resped to ル and will be denoted by y14( が) .
N ow , let 尺 be a Galois eχtension of degree 刀 〇ver ヵ and let /) be a rational prime number prime to 刀. T hen, the fo110wing tw o facts are fundamental in this paper. N amely ;
(1) T he restriction of thenorm map 哉 /4: G ( 瓦) → G ( /)) to 次 ( ♪) is an isomorphism from
y1, (♪) onto (み(♪) . ( cf. Y okoyama 〔4〕)
(2) lf j : G (♪) →G (j ) isthehomomorphism of ♪-dassgroupof んtothat of μ71nducedby
extension of ideaIs, then we have j ( G ( ♪) ) g G ( /)) . ( cf. N akagoshi 〔2〕 )Since j ( G ( Z)) ) 三 次 ( 瓦) we have 次 ( ♪) = j ( G (/・) ) clearly. T hus, considering G ( ♪) as a subgroup of G ( 夕) , we m ay put G ( ♪) = y11( 夕) in our case.
ln this paper we w m deal with the case where 瓦 is a relatively abelian extension of degree M over た whose Ga101s group G ( K yk) is of type ( lj , … j ) , yis a rational prim e number. F or the special case where m = 2 we have proved in my previous paper 〔3〕 the fo110wing theorem.
N amely ; .
TH EOREM A . L en be 皿 α胎 b面 c lmmbey 万d d of 夕池 e degyee, 皿 d ld K be a 畑 a白 幽
岫d au exteMsio芦 of degyee 12 0叱y k 扨hose Gdois F o呻 G ( K 陳 ) is 可 個)e ( /, /) , l iS α
mtiond j)yime m4mbey, L d F 1, 瓦 , …, Fh be the 卸o加y 泌teymd 訟 e万dゐ be棚 em 砲皿 d K .坏 j) is a mtio皿 1 似ime m4mbey 皿 d p≠1, the11 CI! (夕) /次 (ヵ) is 面co観知sd i戒o 甫e diyed
卸O加d ゛ 到lO 面昭 ; Cj j) )/次 ゆ) 二y1バダ)/次 (♪) xAμj)) /次 (♪) ×……y1& 1(♪)/次(カ) ・
Using what we mentioned above, we may replace y14( 加 and ノ1バ:/)) ( f= 1, 2, … , /十 j ) by
G ( 夕) and CR(,1)) ( f= 1, 2, …, /+ 7) , respedively and rewrite above formula as following.
N amely ;
Cj j)̀.)/G(瓦) = C球j)) /G(♪) xC球j) ) /G(♪) ר…゜xC& l(♪)/G(♪)・
N ow , for the class number relation w e have the following theorem immediately.
THE OREM B . ( d . N akagoshi 〔2〕 ) . L d 筧oば iom 皿 d G sMm囲 om be same αs 岫 o叱 , Thm 抑e k ue
Dept. of M ath. , F ac. of Gen. E duc. , Gifu U niv ( Received 10. 12,1987)
22
屁 (夕) 二臨 (坦 臨 (夕) ……hF。 (Z)) / ( 瓦(夕) ) ʼ.
T he purpose of this paper is to extend above tw o theorems to the general case w here ʼ回 > 2 .
2. 0 ur main theorem is as following.
T HEOREM . L d k be 皿 d geb面 c mtmbey μd d of 月面 e degyee, 皿 d ld K be a y出 面 幽 αbd 皿 d a siotl of degyee r olXey k uJhose Galois gyo呻 G ( 尺 μ ) is 可 削 )e ( /, /, … , /) , H S a mtioMd 似imemtmbey, L d F1, F2バ ‥, F t, 1yuhe托 t= ( ? - j ) / ( /- 7) , be tk l)yoj)ey i㎡eymd i-
ate夕dds be励e匹 k aRd K sMd tk t扨ek w 〔Fi : k〕 = 1加y i二l , 2, …, t. lf l) isa mtiolld
1)yime mtmbey 皿 d l) ≠1・ thm C K(ヽ:釣 ZC j 、1) ) is decoml)osd i㎡o the diyed μ odud as 到 lowingOH T A K iichiro
(1)
Since we have
G 々 ) /Cバ 瓦) こ C衣夕) /G (♪) Z Cべ 1)) /G (♪) (j = 1, 2, …, s) ,
we obtain from (4) the decomposition of Cん.( /)) ZCJ ) ) into direct product asC衣 瓦)yCべ 琵) g Cら (♪) ZC臥:/・) x Cぷ 、 1)) ZCJ ) ) ×……x CF。(夕) ZCh(、1)) (5)
d early・Finally for f= 1, 2, … バ the composite fie! d 凡 石 is a relatively abelian extension of degree
Cg(j) ) /G (夕) 二Cべ 琵)/G (♪) X CR(、 琵)/G (夕) χ…… ×CR(j) ) /G (坦 .
M oyeow y, 扨e h 提 tk c加ss mt琲bey ydαtioR
≒(夕)二臨(夕) 似(夕)……hF,(♪) / (/4(j )y-1. (2)
PROOF. W e shall prove our theorem by induction on m . Since w e have both T beorem A and T heorem B for the case where 辨 = 2 , w e may let 琲 > 2 and assume that our assertion is true for 朋 - 7 .
L et L 1, L 2, … , L S, where s= ( M - 1- 7) / ( /- 7) , be the proper intermediate fields betw een
瓦 and 尺 such that wehave 〔Lj: F1〕 = yfor j = l, 2, …, s. Then, by our inductiveassumption Cχ(j ) yC球:ダ) isdecolllposed i趾othedilʻed pl゛odlld 21sfollo゛ i昭 ;
CE(ヽ j) し )ZC球j) ) = Cべ j) ) yC球,1)) x C4 (♪) /Cバ 瓦) ×……x C獄j) ) ノC以j) ) . (3) Now, every乙 (j = 7, 乙 …, s) isarelativelyabelianextensionof degree/20ver k whoseGalois
group G ( Lj Zk) is of type ( 1バ ) , and hencethere are /+ 7 proper interlnediate fieldsbetween ん and L。, one of them must be 瓦 d early. lf we denote them̲ by Fb Fjb 巧 2, … , Fjt( j = 1, 2,・‥, s) , then it followsfrom Theorem A th4t Cら.ひ ) ZCJ ) ) (j = 1, 2, …, s) isdecomposed into
the direct product as follow ing ;CI、(j) ) /G (♪) = CE心) ) /G (夕) x Cら (珀 /G (坦 ×…… x C几(夕) /G (坦 . (4)
/2 0ver ん and hence for sonle j ( /≦に S) we must have 瓦 石 = 乙 evidently. Since i bove 乙 is uniquely determined by 凡 d early and w e have な = / - 7, every 瓦 U ( 1匹 j ≦ t, 1匹 U≦ 1) must appear in { 凡 し = 2, 3, …バ } only once. N ow it is easily verified by (3) and (5) that we have
Cj 、 琵)ZCE心))
On the p-Class Groups of Relatively A belian N umber Fields 23
こ Cバ:夕) /G (ダ) X CR(、 1)) /G (/)) ×…… X C球、 痢)/G (ヵ) .
CE(し虻)/Cバ瓦) g Cj :/ʼ) /G (友) / CR(ヵ) /G (夕)
d early, our assertion ( 1) follows from (6) immediately.
M oreover, if w e consider the order of groups in both sides of ( 1), our last assertion (2) follows clearly. T hus, our theorem is proved completely.
R eferences
〔 1 〕 S. Kuroda, 0 ber dieKlassenzahlen algebraischer Zahlk6rper, N昭oy Mαth. J. 1 ( 1950) 1- 10.
〔 2 〕 N . Nakagoshi, A N oteonX-dassgroupsof certain algebraicnumber fields, j . M g l&e( n eoり 19 ( 1984) 140- 147.
〔 3 〕 K . 0 hta, 0 n thep-dassgroupsof a Galoisnumber field and itssubfieldsj . Math. Socj 砂皿 30 ( 1978) 763- 770.
〔 4 〕 A . Y okoyama, 0 n the relative d ass number of finite algebraic numbur fields, J、Math、Soc. ル μ ,z 19 ( 1967) 179- 185.
(6) Since we have
