Title On the class numbers and the ideal class groups of certainalgebraic number fields

Author(s) OHTA, Kiichiro

Citation [岐阜大学教養部研究報告] vol.[25] p.[57]-[67]

Issue Date 1989

Rights

Version 岐阜大学教養部 (Dep. of Math., Fac. of Gene. Educ., GifuUniv.)

URL http://hdl.handle.net/20.500.12099/47723

※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 theclassnumbers and the ideal elass groupsofcertain algebraic number fields

57

K iichiro OHTA

§1. 1ntroduction.

ln genera1, 1et たbeanalgebraicnumber fieldof finitedegree. Then, theabsoluteideal class

groupandtheclassntlmberofカwillbedenotedbyG and瓦 respectively. Next, foranyprime

number♪the♪-Sylow subgroupof (心will becalledthe夕Xdassgroupof ルandwm bedenoted

byら (夕) , whoseorderwillbedenotedby瓦(夕) . Moreover, let瓦 beaGaloisextensionoffinite

degree over ん. Then, the subgroup of all ideal classes of G (瓦) which are ambigous with

respect to K 陳 w ill be called the ambigous y class group of 尺 with respect to A・ and wm be

denoted by y1衣:♪) .

Now, let 尺 bea Galoisextension of degree 刀over ヵ and let ヵ be aprimenumber prime

to が. Then, the following two facts are fundamental in thispaper. Namely ;

(1) Therestridionof thenormmapNIい : G (♪) → G (♪) toy11(♪) isanisomorphism from

凡 (夕) ontoG (夕) . (d. Yokoyama [10D

(2) lfj : G (夕)→G ひ) isthehomomorphismofy dassgroupofんtothatof尺 inducedby

extension of ideals, then wehavej ( G (♪) ) ΞG (♪) . ( cf. N akagoshi [4] )

Since j ( G (夕) ) ⊆ノ14(が) wehave几 (瓦) = j ( G (♪) ) clearly. Thus, considering G (♪) as a

subgroup of G (♪) , wemay put G (瓦) = y14(瓦) in our case・

ln this paper, first, weshan deal with the casewhere尺 is a relatively abelian extension

of degree戸 over ヵ whoseGaloisgroup G (K yk) isof type ( 口 , … , /) バ isaprimenumber. For

the special casewhere 辨 = 2 wehave proved in [5] the following theorem. N amely ;

T HEOREM A . Leほ be回 向 eb屈 cm ・mbey徊 ld of 鋤 itedegyee, 回 d let K bea y血 白 幽

abdiall a tetlsioR of degyeep oley k uXhoseGdoisgγo呻 G ( K Zk) 包oj` 節 e ( 口 ) j is ay ime

H s & れ £ d F1, 瓦 ,… , F tu betk 但o佃y inteymd 斌 ej dds be励 ea k n d K 、 町 い sα詐面 e

筒柑 ≠1, tke11 C11(夕) / 几 (夕) {s decoml}osd 緬to tk diyed y o加 d as加Uo切緬g ;

G (瓦)/ 瓦 (♪) = y1バ:♪) / y14(夕) xy1衣♪) / 凡 (♪) ×…xylh (Z)) / 凡 (♪) .

Using what wementioned above, wemay replaceノ11(♪) and y1ぶ :蕉) ( f= 1, 2,… , /+ 1) by

G (夕) andGべ夕) ( f= 1, 2,…丿+ 1) respedivelyandrとwriteaboveformulaasfollowing ; (cf.

Nakagoshi 圃 )G (♪) / G (夕) = GX♪) / G (が) xC乱(夕) / G (夕) ×…xG。(♪) / G (夕) .

The first aim of this paper is to extend above theorem to thegeneral casewhere 琲≧2.

Namely, in§2 weshall provethefollowing theorem.

T HEOREM X、Let k be回 向 eb面 clmmbeyμdd of 畑 治 degyee, 回 d ld K bea 畑 a面 幽

abdiaRate肴sio肴of degyeer olXeyk uJhoseGdoisgγo呻 G (K陳) zsが ㈲g( 1バ,…バ)バ 后

Dep. of M ath., Fac. of Gene. Educ., Gifu Univ

K iichiro OHTA58

THEOREM 2. Ld K be n abd n exte, lsion 可 degyee ? ‘ ow r Q uJhose Galois gyo呻

G (K ZQ) isof 幼e (2, 2,…, 2) . Ld k , k2,…, 島, g加q X= 2 - 1, bethedi加y四t q皿dmlic

sMbβeldsof K、 jy扨eαs誂観eG1こGi(2) ×民 and(≒こCバ2) ×Bi釦yi= 1, 2,…, /, 法四 回 加回

民 = 既 ×瓦 ×…×μ (diyed 加o血d) .

Moyeow,・, が lhe山ssumbey廠of K isodd, tha da oh gtheoyde7 0j Bj )yb沃仁 1, 2, …。 )

wehave

hχコbl硲‥砿

Now, it follows immediately from abovetheorem that if weknow theclassnumber 垢 of

瓦 is odd, then we are able to determine the class number and the idei l class group of 瓦

explicitely by considering only theqUadraticsubfields of 瓦 Hence, in §3 weshall investigate

to construct the number field 瓦 whose classlnumer is odd foT 枇 = 2, 3 and 4.

Finally, in§4weshallgivesomenumericaleχmplesandlisttherealnumberfieldswithodd

dass number in a table for 附 = 2, 3 and 4.

§2. Proof of Theorem l and2.

First, werecall thefollowing definitionandtheorem discussedin [51, whichareimportant

tools to prove Theorem 1・

DEFIN汀ONべd. [5] ) Leは ben α胎わ咄cu mbeyj eldoj`ji咄edegyee皿d ld K わeaG山 is d a sion of d昭yee mn opey k Th回 K 戒 1日 )e cd d 服 (y1) -ext四 s1011 0wy 12 1f tk

GdOiS舒O呻 G= G (瓦μ) sd 球es服 到 Io面昭 co姐伍0 ;

(A) Gkαsα肴oymd s油訂o砂 N of oyぬynalld筧s油乎o卯sH1, H2,…, Hn可 sαmeoydey

m swk tk t we随従

GこNH1こNH2こ …こNHnα祖 ∩瓦 = 佃} 力r f≠j, 抑陥犯 側edmotebyE匝eM戒tdeme戒

Of G.

T HEOREM B. ( cf. [5D Leは ben α胎 b面 c m4mbey j μd of j 油 ed昭yee回 心 d K ben

(A) -at匹sio肴of degγeemR o叱y k Ld F, L1,L2, …・玩 be服 s油βddsof K con刈)回di昭

托s佃c面dy to tkes油gyo呻sN, H1, 瓦 ,…, 瓦 Of 加 GαlOiSF O呻 G (K Z& ) bjy伍eGdoistk o巧.

lミf p isa l)γlme侃m

α鈴imem4琲bey. Ld F1, F2,…, Ft, 抑陥托 tこ ( μ - 1) / ( /- 1) , b八 陥 鯉o加y 面 eymed咄 e万出 s

be訟 em k alld K s14ch tk t we随 叱 [死 丿列 = /力y X= 1, 2,… , t, が p is αp召琲emtmbey alld

♪≠I, tk筧C11(が)/G (瓦) is dea)m知sd 池 o加 d前d 卸o加d αs到 lo戒昭 ;

G (♪) /G (瓦)= Gバ:瓦)/G (瓦) xCh(j) ) /G (φ) ×…xC衣j) ) /G (j ) .

Moyeopey, u)e k w tk dαss mtmbeγ?・eld ion αs知110面 14g ;x

臨(友)= (皿恥心)) ) /(/4(yヴー1。沁 1

Neχt, let 尺 be an abelian eχtension of degree2・ over therational number field Q whose

Galoisgroup G (尺/Q) isof type (2, 2,…, 2) . T . Kリbota [31, H . W ada [7] andH . Cohn [1]

studied the class number of 瓦 by using its unit group and calculated several numerical

examples for thecasewhere辨 = 2 and 3. 0 n theother hand K . Uchida [6] investigated the

casewhere瓦 isimaginary andobtained almost all of such瓦 whoseclassnumber isone.

Thesecond aim of thispaper istostudy theclassnumber andtheideal classgroupof such

number field 瓦 asdescribed abovewithout using itsunit group. N amely, asan application of

Theorem l we shall also prove in §2 the following theorem.

59On theclassnumbers and theideal classgroupsof certain algebraicnumber fields

G O) ZA心) ) = ylr(夕) /凡 (瓦)×< ノ1h(が) , 竃 ,(♪) , ・。; , y11. (瓦) > /ノ1衣:夕) .

Here, fronl sanle reason asbefore wemay replace凡 (♪) , ム (夕) α加 甫 か ) ( 汗 1, 2,… ,

疋) by C心) ) , G (夕) and G ,(♪) ( f= 1, 2,…, g) respectively and rewrite above formula as

f0110wing ;

G (♪) /G (夕) = G (夕) /G (瓦)×< G j :♪) , Gズ♪) ,…, (‰(♪) > /G (♪) .

PROOF of T HEOREM 1. W e shall prove our theorem by induction on 辨. Since we have

Theorem A for thecasewhere 辨 = 2, wemay let 撰 > 2 and assumethat our assertion istrue

for 附 - 1.

Let F beany oneof 瓦 ( 1≦f≦O and fix itat atime, W edenotetheGaloisgroup G (K ZF )

by χ Then Ⅳ is an abelian group of order 戸一1and of type ( 1バ ,‥・バ ) . N ext, 1et L1, L2,… ,

£t, whereX7 ( 政 - 1) / ( Z- 1) , betheintermediatefieldsbetweenんand尺 suchthatwehave [乙

丿刎= / l for j= 1, 2,…バ. Sinceonlys= ( /’-1- 1) /(X- 1) £fwehaveF⊂乙⊂瓦 itfollows

immediately that there exist exact g= 乙- s= M-1 £j for which we have F n乙 = ん. For

conveniencewemay denotesuch g 句 by L1, L2,… , Lzj respectiveny. M oreover, wedenotethe

GaloisgroupsG ( K Zk) and G ( K yL5) (j = 1, 2,… , め by G and罵 (j = 1, 2,… , u) respectively.

SincewehaveF n乙 = んfor y= 1, 2,… , g and Lj、Ljl= 尺 for j1≠ふ it fonowsimmediately that

wehaveGこN111= N瓦 = …= jV7亀and瓦丿 瓦, = { ε} forjl≠ふ wherewedenotebyEtheunit

element of G. Hence, considering theorder oI N and瓦 (j = 1, 2,… , め resped ively, it follows

easily that 尺 is an (A ) -extension of degree μ = & over ん. Applying Theorem B to our case

wehave

(2. 1) Cx(夕) /G (♪) = G (♪) /ら (♪) ×< Q (夕) , …, Cし(♪) > /G (夕)

dearly.

N ext, 1et F ’ bealso any oneof 瓦 ( 1≦ f≦O andweassume F ’≠F . Then, since尺 is an

abelian eχtension of degree 戸一20ver FF ’ whoseGaloisgroupisof type ( 口 , …j ) , it iseasily

seen that thereexist only z7= ( /’-2- 1) /( Z- 1) 乙 。佃 < j ≦O such that wehaveFF’ ⊂ ち .

Thus, for exact s- zノ= 戸一2乙 ( 1≦j ≦砥) wehaveF ’ ⊂乙 and henceC副 ,1) ) ⊆ (気心) ) clearly. lf

weassumeF = 瓦, then, aswemay let F ’ beanyoneof 瓦 withy≠jl wehave

ら1(♪)…G。(x,) CFiU(,1) )…ら ,⊆< CE,(♪),・へGX夕) >

and hence

(2. 2) G X夕) …Cyブ j) ) G 。 (夕) …CJ 、1) ) /G (瓦)

⊆< Gズダ) ,…, G刀)) > /G (♪)

dearly.

0 n theother hand, if wedenotesquadraticsubfieldsof ん byFjb Fj2, … , 瓦μ or j = 1, 2,… ,

zも then from our inductive assumption we have

CJ j) ) /G (♪) こC恥(,1) ) /G (瓦)×CJ ミj) ) /G (/)) ×…×(≒ (ダ)/G (♪) (j = 1, 2,…, め

and this implies

CJミj) ) /G (瓦) = Cぶ ,1)ス) G衣ダ)…C5 (が)/G (♪) (j = 1, 2, …, め

immediately. Sincewehave瓦 ∩烏 = ん(1≦j ≦u) by our assumption, it follows easily

(‰(♪) /G (が)E (ふ(♪) …Gい(夕) C5 、(♪) …G ,(/)) /G (夕)

for j = 1,2,… , zf, and hence wehave from ( 2. 2)

(ふ(♪) …G ,。(夕) Gバ:夕) …Gか ) /G (夕) = < Q , (夕) ,…,CU(少) > /G (夕)

dearly. Now, from (2. 1) and aboveformulaweobtain

(direct product) .G=几G(♪)

60 K iichiroOHTA

G O) /G (夕) = Cバj) ) /G (瓦)xGエ。)…C4 心) ) G。(♪) …Cj しj) ) ZCJ ) ) .

As瓦 isanyoneof Zquadraticsubfieldsof瓦 thisformulaimpliesour assertionistrue, that

is, G (♪) /G (瓦) isdecomposedintothedirectproductasfollowing ;

G (瓦)/G (友) = Ch(加/G (φ) xC以j) ) /G (瓦)×…xC球j) ) /G (j ) .Thus, 0ur theorem isproved completely.

PROOF of T HEOREM 2. 1ngeneral, letルbeanalgebraicnumber fieldof finited6gree. Then,

it is well knownthat we have

Hence, to proveour theorem it issufficient to show that for any oddprimenumber カ wehave

ln addition to Theorem 2 we shall mention to the special case where we have 臨 = 1.

N amely ;

COROLLARY, No厭 iolls aRd αssR琲誹i匹 s be飢g sαme as Tk o犯琲 2 皿 d mo犯o叱y we

αssume加 山 ss u mbey 厦 oチ K is odd, Tha 回 k ve恥 こ1 1f n d only 汀 b1こ b2= …みt= 1.

PROOF. This coronary follows immediately from Theorem 2.

§3. 0n certainnumber fieldswithoddclassnumber.

ln this sectionweshall investigatetheconcretecasewheretheclassnumber iscalculated

explicitely by Theorem 2. N amely, weshall construct thenumber fieldsof degree4, 8 and 16

whose classnumbers are odd.

㈲ Thebiquadraticcase.

First weshan prove the following theorem.

THEOREM 3. 1 d X = Q ( √y , √凪) be α biq皿 dm励 肴umbey j dd, uJheye l) is a l)yime

゛“” lbet o d “ is “ s卯 ゛ e一介ee泌t昭ey無品 d o匁 Ld ねこ Q ( , /万 ) beαqu dmticsttbβdd of

K. Tha , がK is皿 皿mm垣d d a si匹 o叱y lこ皿d四 随従只 臨, 伍a hJ so心 ,

PROOF. W e assume our assertion is not true. Then we have 臨 = 2・・& with ,x≧1 and

(2, 4 ) = 1. LetL bethesubfieldoftheabsoluteclassfieldof五7suchthat£ ⊂£ and [1 : f ] =

2”. Thむn, it is easily verified that £ is a Galois eχtension of degree 2”+l over Q. Since the

Galois group G二G (LノQ ) has the principal series as a 2-group, refining the normal series

G⊃G (K/Q)⊃ { 吋 , wherewedenotebyEtheunitelementof G, weknow theexistenceof

normal subgroup 耳 of G such that G ( K ZQ ) ⊂亙 ⊂G and [H :・G (K ZQ ) ] = 2, Let F bethe

subfield of £ corresponding to H by theGalois theory. Then, F is Galois over Q because耳

is normal in G. M oreover, it is easily verified that F is an unramified extension of degree 4

0ver ゑ As thegroup of order 4 1s abelian, it follows immediately that F is contained in the

absolute class field of 尨 Hence, 瓦 must bedivided by 4. This is a contradiction clearly.

THEOREM. 4, Ld K こ Q ( √瓦 √R ) be α biq回 dyd c u mbey j dd, uJheye l) is a l)yime

刄umbey αud q is 誼 k y eq皿 l to -1 0y α j)Mme 筒m (mod 4) (叩1d q≠1) .

Mo犯owy, び j) isoddバhm weαssu琲eか三3 (mod4) a j 加 三5 (mod8) . Ld lこ三Q ( √瓦 )

be 服 q回 dyatic s福βdd of K . Tka , が 回 αssume 臨 is oddバha k is d o odd,

To prove this theorem weneed the following lemma due to K . lwasawa [21.

LEMMA 1. ( cf. lwasawa [2] ) Leは be朋 面鋤 臨 c lmmbey j dd of 夕咄 e 面gyee n d ld

(direct product) 。G (が) こCJ j) ) ×(乱(瓦)×…XG ,(瓦)

But, this is an immediate result applying Theorem l to our case.

61On theclassnumbers and the ideal classgroupsof certain algebraicnumber fields

K beα11 eχtmsio筒 of degyee l) o叱y k, wk ye l) is α餅 imemtmbey. Moyeo叱y, 扨eassMmetk t

tk yed sts on砂 os 函 med屈 soy of k which is mm示砲 泌 K . Tka , 仔 hJ snot 函 isible by

飢 th匹 厦 isalso筒otd屈s伽ebyl) .

PROOF of THEOREM 4.

(1) Thecase♪= 2, Theprimeidea1(2) of Q istotally ramifiedin尺 andhencewehave

(2)= p2 1n 瓦 wherepisaprimedivisor of 尨 Moreover it it easily verified thatpistheonly

prime divisor of カ which is ramified in 尺 As 砥 is odd by our assumption, applying Lemma

l to our case our assertion follows immediately.

(2) ThecaseZ・ミ 3 (mod 4 ) and加 三5 (mod8) . Theprimeideal (2) of Q isdecomposed

into prime divisors of K as following ;

(2)= p2, 靖 /(汐= (2)2.

Sincetheideal (2) remainsasaprimein ゐ, it iseasily seen that (2) istheonly primedivisor of

んwhich is ramified is尺 Now, applying Lとmma l our assertion follows imediately.

T HEOREM 5. L d K = Q ( √j , √J ) be α biq皿 dM& 肴Rmbeγ斤eld, 扨k γe l) is 皿 odd

y而別,? がg辨加y回j αfsα将忽切一力認 fが偕臼・♪パ辨・? 必♪. 訂θy召θ誂?y z47召αssμ7μ召請厨 勿召加叱

(j)=-1α心α三3(mod4). Ldk=Q(√J) beαq皿dmtics油μddofK. Tha,ijT回l sume臨 isodd, then hs isd oodd.

PROOF. Aswehave( j ) = - 1 by our assumption, theprimeideal (φ) が Q remainsasa

prime in んatld it isramified in尺 clearly. 0 n theorher hand theprimeideal (2) of Q isramified

in ルbecausewehaveαミ 3 (mod 4) by our assumption. M oreover, if ♪( L (mod 4) , then, the

idea1(2) isunramified in Q ( √j ) . lf 夕三3 (mod4) , thenwehave砂 三1 (mod4) andhence

theideal (2) isunramified in Q ( y万 ) . From thesefactsit iseasily seen that theprimedivisor

of (2) inA・isunramifiedin瓦 Hence (♪) istheonlyprimedivisor of んwhichisramifiedin瓦

Now, 0ur assertion f0110ws immediately from Lemma 1.

T HEOREM ら. Ld K = Q ( ぶ , √万) be 皿 im昭i皿 砂 biq皿 dm良 筒Rmbey j dd, 扨keye j)

皿 d q aye diがeye戒 odd l)yimemtmbeys st4ck tk t l) ミ q三3 (mod 4) . Let k= Q ( 刀海) be the

q皿dmtic su吊 dd of K 、 Then , if weαssumehJ s o服 , tka k is 幽 o odd.

PROOF. Sincewe have知 三 付) 三 - 9ミ 1 (mod 4) from our assumption, the prime ideal (2)

of Q isunramified in 瓦 Hence, it is easily seen that all finiteprimesof ヵ areunramified in

瓦

N ow, weassunleour assertion isnottrue. T hen, it followsby thesameway asinTheorem

3 that thereexists an unramified extension of degree 2 0ver 尺 such that it is alsoGalois over

Q. Wedenoteitby£. LetF bethemaximal real subfieldof£. Then, itfollowsimmediately

that F isanunramifiedextensionof degree2 0ver 尨 Thisimpliesthat 瓦 isdivisibleby2. This

is a contradiction clearly. Thus, our theorem is proved completely.

(b) Theocticcase.

First weshall provethe following theorem. N amely ;

T HEOREM 7. Let K = Q ( √2 , √j , √i ) be 皿 od cμdd whose G山 is F o呻 {s 原 図 }e

(2, 2, 2) , wheyel) isd heyeq皿にo - 10y皿 odd l)百me筒m (mod4) α肩

qis 皿 odd 函 meu mber d胎 ye戒 斥om 払 Moyeo斑 冶k y加 ミ 5 0y qミ 5 (mod

8) . Let k= Q ( ,/y, √i ) betkebiq皿dmtic誹吊ddof K . Thm, if 扨eassRmekμsodd, th匹

k iSd OOdd.

PROOF. First it is easily seen that the prime ideal (2) of Q is totally ramified in

Q( √2 , √瓦). Buttheideal(2) remainsasaprimeeitherinQ ( 痢 ) の・加Q( √ぷ) according

toヵ(7三5 0r ・7三5 (mod 8) . Henceit followsimmediately that ideal (2) isdecomposed intothe

prime divisors in 瓦 as following ;

(2)= 畢4, yVk/。畢= (2)2.1f wedenoteby p theprimedivisor of (2) in ん, then wehave2= 岬, Λ1/。抑= (2)2clearly. Thus,

it is easily verified that p is the only prime divisor of ゐ which is ramified iil 瓦 Applying

Lemma l to our case our assertion follows at once.

Now, to proveneχt theoremsweshall need the following lemma. N amely ;

LEMMA 2、 Ld & be 皿 dgeb咄 c mtmber 加 は of β池 e degyee 皿 d let K be a Gdois

a tm sioll of 面g粍e 8 0叱y k、 Ld F be 朋 i戒eymd ide j dd be励 eeu lこ α11d K sMch tk t

[F : 列 = 4皿dtheGdois訂o呻 G (FZk) 后げ 帥ぺ2, 2) . Thm, が伍e粍existsαj)言md面d

pげ ヵ a油 必zX四 加q q= P2バVE/丿 = q2加 石 g加g P isα函mH dd of F wk H s

mlyα琲浜ed泌 K, tk11K isαuαbdiα筒ate肴sioMowy k、

PROOF. AstheGa101sgroup G = G ( K Zk) isof order 8, toproveour lemmait issufficient

to show that neither thequaternion group nor thedihedral group of order 8 1s isomorphic to G.

Since theGaIoisgroup G ( F μ ) isof type ( 2, 2) by our assumption, thereexist 3 proper

intermediate fieldsbetween ヵ and F . W edenotethem by £ l, £2 a姐 £3 respectively. Then, it

is easily verified from our assumption that p is not decomposed into theproduct of different

prime ideaIs in any L i (しi = 1, 2, 3) .

N ow, we shall consider thedecomposition of p into the primedivisors of 瓦 Then, from

our assumptionwehaveeither p= 雫2, yVk/h雫= p40r p二(雫1叩2) 2j V臨 畢f= p2clearly. lf we

have the first case, then wedenote the inertial field of 畢 by 7こ Then, it fo110ws immediately

that theGaloisgroup G ( T ノk) is a cyclicgroup of order 4, that is, 7` is a cyclicextension of

degree4 0ver 瓦 。 0ntheother handif wehavethelatter case, thenwedenotethedecomposition

field of 畢l by 乙 . Then, it iseasily verified that 乙 is an extension of degree 2 0ver んwhich

isdifferent from any Lバ ,i = 1, 2, 3) . Thisimpliesthatthereexist at least 4 proper intermediate

fields between ル and 尺 each of which is an extension of degree 2 0ver 尨

Now, we assume G is isomorphic to eiter the quatem ion group or the dihedral group of

order 8. Then, it f0110wsimmediately that thereexistsnocyclicextension of degree 4 0ver ん

and there exist only 3 proper intermediate fields between ん and 尺 each of which is an

extension of degree2 0ver 力. This isa contradiction clearly. H ence, G mustbeabeliangroup。

THEOREM 8, Ld K こ Q ( √2 , √j T, √i ) be 皿 od c j dd w加 se Gdois F o砂 is 可 幼 e

(2, 2, 2) , Wk yej) isdtk y 同心にo - 10y皿 odd j)百観em4mbeysttch tk t j) ミ

62 K iichiro OHTA

ん干Q ( √が, √i ) belhebiqudmtics14球eldof K. Tha , が回 ass14me玩 isαlSO Odd、

PROOF. VVe assunlle our assertion is not true. Then, it follows by the same way as in

Theorem 3 that there eχists an unramified extension of degree2 0ver 瓦 which isGaloisover

Q. Wedenoteitby £ .

AswehaveG ) = 1 from our assumption, theprimeideal ( g) of Q splitsin O ( √D /and

wehave (g) = pl芯 g加g ㈲ :に 1, 2) isaprimeideal of Q ( √j T) . Moreover, theideal (瓦)

od4) a j

| =1. £dtk n hバ s

・7 1s n o習 か鋤 g u m細 ・ j i加 ya l か s Z・. jg ∂ygθz7召y z4夕g αsszx777g G 卜 11_ 八 y m m χ 1 . 1 1 ・ . . . . . . . . . _ _ - = - - g

ideal of 尺 such that wehavejVkyQ( √が) 0 1= q12, SinceO l isunramifiedj n L , applying Lemma

2 to our case, it follows immediately that £ is an abelian extension of degree8 0ver Q ( √j ) .

0 n the other hand, aswehave匹 3 (mod 4) from our assumption, wemay put (2) = p2 1n

Q ( √y) , wherepisaprimeidealinQ ( √j ) , Moyeowy, astk ided (2) isltllmm涵ddthey

加 Q ( √万) or in Q ( 刀海) accordingto9三1 0r 9三3 (mod4) , it iseasilyseenthat pis

unramified in ん. But, sincetheideal (2) istotally ramified in Q ( √2 , √j ) , it f0110wsthat the

prime divisor of p in んmust be ramified is 尺.

N ext, sincetheGa101sgroup G ( L yQ ( √7F) ) isabelian, it iseasily verifiedthat theinertial

field of any primedivisor of p in £ with respect to LZQ ( √j ) is uniquely determined. W e

denoto it by 7こ As theprimedivisor of p in 瓦 isunramified in £ , it fonowseasily that、T is

aproper intermediatefieldbetweenヵ and£ andwehave尺≠乙 Asanyprimeideal ofんwhich

doesnot divide p isunramified in £ , it f0110ws at oncethat T is an unramified eχtension of

degree 2 0ver た This impliesthat 瓦 isdivisibleby 2. This isa contradiction clearly. Thus,

0 ur theorem is pr ov ed com pletely .

T HEOREM 9. Let K = Q (: √万, √万 , √y) be皿 oc励 μdd, tりk yel) is d服 y eq皿 に o - 1

0y α11 0 dd l) yim e 肴m

j乙

gぐjl

仁y

On theclassnumbers and the ideal classgroups of certain algebraicnumber fields 63

is ramified in Q ( √i ) and ramains as a prime in Q ( √2 ) becausu wehaveG ) こ - 1 from

our assumption. Henceit iseasily verified thatwehave q1= こ 121n瓦 where£ l isaprime

糾観is oddバ hm抑e αssR衿1e

PROOR W e assurne our assertion is not true. Then, it follows by the same way as in

Theorem 3 that there eχists an unramified eχtension of degree2 0ver 尺 which is Galois over

Q. W edenoteitby£ .

SincewehaveG ) = 1 from our assumption, theprime ideal ( Q) げ Q splits in Q ( √j )

and we may put ( 3) = ql q2, W zm ・ q1 α心 q2 αg j i砂 gが 夕可大 池 詰 みz Q ( y j ) .

M oreover aswehave( £ ) = - 1 from our assumption/it folloJ seasily that qf ( i= 1, 2) remains

asa primein Q ( √j ノ √F) . qf ( j = 1, 2) isramified in瓦 dearly andhencewehaVe ql=/

a12, 篤/, ( √j ) ら1= q12, wherea l isaprimeideal of尺 whichisunramifiedin£. Now,applying Lemma 2 to our case, it follows immediately that the Galoisgroup G ( £ / Q ( √j ) )

is abelian.

On the other hand, from our assumption for 夕, ・7 and y it is easily seen that the prime

divisor of ideal (2) in ん are unramified in 尺 . M oreover, as we have(夕) = - 1 from our

assumption, theprime ideal (r) of Q remainsasaprimein Q ( √y ) and it splits inゑ Hence

wemayput (y) = rl r2, where tl and r2aredifferent primeidealsof ヵ whichareramified

in 瓦

N ext, sincetheGaloisgroup G ( LyQ ( 巧 ) ) isabelian, it iseasily verifiedthat theinertial

field of any primedivisor of ( y) in £ with resped toLyQ ( √j ) isuniquely determined. W e

denote itbyT oThen, it followseasilyby thesamewayasinTheorem 8that T isanunramified

extension of degree2 0ver ゐandhence瓦 mustbedividedby 2. T his isa contradiction clearly.

Thus, our theorem is proved completely.

(mod 4) , 肋a u 。x吋 加叱 に 1 (mod4) . Mo托owy扨eαss14me

Ld たこQ ( √j5, √y) be服 biq回d咄ics油β出 of K. Tha

厦 iSdS0 0d1

= 1 α77j = - 1.

64 K iichiro OHTA

(C) Thereal biquarticcase.

Now, we shall prove the following theorem. N amely ;

THEOREM 10. Ld K = Q ( √2 , √j , √肌 √F) be a yeαl mlmbey j dd of degyee 16

whoseGdoisgyo呻 is of 削)e収 , 2 , 2, 2) , wherep, qandr aredifferentoddprimenumberssuch

that we have p三r三3 (mod 4) , ・7こ 5 (mod 8) , ( 夕) = 1 α心 (夕) = G ) = G ) = ぐL. £ d ヵ=

Q ( √j , √i ) 加 法g加卿而忽面 々所dj が瓦 刀za, び回 心ggg瓦八心必法四 尻 £sα励

Odd、

PROOF. W eassunleour assertion isnot true. Then, it followsasbeforethat thereexists

anunramifiedextension of degree2 0ver 瓦 which isGaloisover Q. W edenoteitby £ .

As we have Z・三y三3 (mod 4) and 9三5 (mod 8) from our assumption, applying the

quadraticreciprocity law to(夕) = 1 and (ク) = 1 1t follows(チ) 二 ( ミル T l immediately H ence,

the prime ideal (瓦) of Q is decomposed completely in Q ( 而 , だ) α11d we 観砂 1)耐 (p) =

り 山 ら , where 喝 is a prime ideal of Q ( ,/万 , √F) . Since wehave( D = -l fronl our

assumption, 則 remainsasaprimein Q ( √2 , √万, √F) . As 、pl isramifiedinK, it iseasily

seen that we have p1二雫12, 裁 /。( √万。芦 ) 雫1= 貼 2, where 剔 is a prime ideal of 尺 which is

unramified in L . N ow, applying Lemma 2 to our case, it follows immediately that theGalois

group G (LZQ ( √示, √F) ) isabelian.

On theother hand, it iseasily verified by our assumption that theprimeideal (2) of Q is

decomposed into primedivisors in Q ( ,/万 , √7 ) as following ;

(2)= q2, 蹟 (√i, √7)/Q q= (2)2

M oreover, it fonowseasily thatqsplits in Q ( √が, √y, √F) becausethe inertial degree of

(2) with respect to Q ( √j , √iy, √F) /Q isequal to2. 0n theother hand, theideal (2) is

totally ramified in Q ( √2 , √F) , and henceramification index of (2) with respect to瓦 / Q is

at least 4. T hen, the prime divisor of q in Q ( √jj’, √i , √ F ) must be ramified in 瓦 ・

Next, since theGaloisgroup G (LZQ ( √y, √F) y isabelian, it iseasily verified that the

inertial field of any prime divisor of q in l with resped to LZQ ( √y, √F) is uniquely

determined. W edenoteitby T 、 Then, it followseasily by thesameway asinTheorem 8 that

T isanunramified extension of degree2 0ver Q ( √j , √4 , √F) andhencetheclassnumber

of Q ( √j , √y, √F) mustbedividedby2. But, sinceitiseasilyseenthatall assumption

of Theorem garesatisfiedinour case, it followsimmediatelythattheclassnumberof Q ( √j ,

√i , √F) isodd. Thisisacontradictionclearly. Thus, ourtheoremisprovedcompletely.

§4. Examples.

Now, we denote the class numbers of Q ( √j ) and Q ( √j , √辺) by yz ( √j T) and

yz( √瓦 √4) respectivelyinthefollowingexmples.

(1) K二Q ( √5 , μ7 ) . Wehaveyz( √副) = k ( μ7) = 1 andyz( √器5) = 6. Hencefrom

Theorem 3 wehuve 臨 = 3.

(2) 瓦= Q ( 1, √面r) . Wehaveyz( √i) = み( √皿[ ] = 1 andyz( √顎T) = 14. Hencefrom

Theorem 3 we have 厦 = 7.

(3) £ = Q ( √T , μ r) . Wehave71- 3 (mod4)丿 ( √2 ) = yz( J T) = 1andyz( √Ry) = 3.

Hence from Theorem 4 we have 臨 = 3.

(4卜 £ = Q ( y7 , √B [ ] . Wehave151三3 (mod 4) , 453三5 (mod 8) and yz( √1 ) =

wehaveゐ( √2) = 1, yz( ,/26) = 2 andyz(

(13) 尺= Q ( √3 , √lヨF, √ΓΥ) . Wehave(

=6. H ence from Theorem 8 wehave 臨 = 3.

(jy)=(サム万一1andyz(√5,√ri)=1fromalid yz( 15) = yz( √B S ) = 2. HencefromTheorem 3. N ext we have yz ( √3 ) = yz (

Theorem g wehave yzχ= 1.

On theclassnumbers and theideal classgroups of certain algebraicnumber fields 65

yz( √mでr) = yz( √B3 = 1. HencefromTheorem4wehave尻こ1.

(5) 尺= Q ( f, yR ) . Wehave83三3 (mod4) , -83三5 (mod8)丿 ( √T) = yz( J 3) = 1 and

yz( √二羽 ) = 3. HencefromTheorem4wehave臨= 3.

(6) X = Q ( √y, √胚F) . Wehave439 - 3(mod4) , (響 ) = -1丿 ( √r) = yz( y剪惣) = 1and

yz( √認9) = 5. HencefromTheorem5wehave尻= 5.

(7) 尺= Q( j, √2213) . Wehave(忌 ) = -1j z( √i) = 1, yz( √2酉 ) = 3andゐ( √223) = 7.

Hence from Theorem 5 wehave 尻 = 21.

(8) 尺= Q ( √3 も ぷ1お) . Wehave23- 3 (mod4) , み( y弱) = み( √3) = 1andyz( √二23) =

3. Hence from Theorem 6 we have 尻 = 3.

(9) 瓦= Q ( √2 √5 , J 9) . Firstwehaveyz( √15, √[9] = 1 fromTheorem3. Nextwe

have yz ( √亙) = yz( y盾 ) = 1 四 j yz ( √皿 ) = yz ( √n O) = 2. Hence from Theorem 7 wehave

尻= 1.

(10) X = Q (i, √2 , √mi ) . Firstwehaveyz( √T, √T皿 = 7fromabove(2). Nextwehave

yz( √r ) = yz( √2) = 1, yz( √涯2) = 2andyz( √2屁) = 6. HencefromTheorem7wehave瓦=

21.

(11) K = Q ( √2 ’ √5’ √ri) ゜ We h e(D 二 1’(廿) 二1 311d み ( √5’ √n:) 二1 fl°0111

Theorem 3. N ext wehave yz ( J 2) = 1 and yz ( J {} ) = yz ( √i n) ) = 2. H ence from Theorem

8 wehave 尻 = 3.

㈲翼=Q(と√2, √r3).Wehave(il)⊃1,(且)=1andノz(√i, yn)=1from[61.Next

(14) j

l (mod

and yz(

宍=1,(yF)=シχJA?=-1andみ(√iyn)=1from-11) = 1, み( -143) = 10. HencefromTheorem gwe

㈲ 瓦 = Q ( j, 、/Ti , 、yr3) . W ehave

above(12). Nextwehaveyz( μΥ) = yz(

have臨= 5.

㈲ 尺 = Q ( √2 , √3‘, √7 , √口) . W eputX・= 3, ・7= 13 and ,・= 7 1nTheorem 10. Thenwe

have(il)=1,(でih)=(j)=-1andゐ(√3, √r3)=1fromTheorem3. Nextitiseasilyseenthat there exists no quadratic subfield of 尺 whose class number is divided by an odd prime

number. Hence from Theorem 10 wehave臨 = 1.

Finally, we shaII list in a table the classnumbersof real number fieldssuch that we can

obtais as the results of theorems is §3 foT j) , q, γ, α≦31. W e m6an ( j) , q, y) a field

Q ( √j , √y, √F) inthebelow. ノ

(1) Theresultsof Theorem 3.

(a) Thefieldswithclassnumber 1.

(2, 5) , (2, 13) , (2, 17) , (2, 21) , (2, 29) , (3, 5) , (3, 13) , (3, 17) , (3, 29) , (5, 7) , (5, 11) , ( 5, 13) ,

(5, 14) , (5, 17) *, (5, 19) , (5, 21) , (5, 22) , (5, 23) , (5, 31) , (6, 13) , (6, 17) , (6, 29) , (7, 13) , (7, 17) ,

(7, 29) , (11, 13) , (11, 17) , ( 11, 29) , (13, 14) , (13, 17) , (13, 19) , (13, 21) , (13, 22) , ( 13, 23) , ( 13, 29) ,

(13, 31) , (14, 17) , (14, 29) , (17, 21) , (17, 22) , (17, 23) , (17, 29) , (17, 31) , (19, 29) , (21, 29) ,

) =-1. Nextwehaveみ(y37)=1,み(√苓iT)=み(√629)=2from Theorem g wehave 尻 = 1.= 4.

K iichiro OHTA66

(22, 29) , (23, 29)

(b) The fieldwithclassnumber 3, (29, 31)

(2) The resultsof Theorem 4.

The fields with class number 1.

(2, 3) **, (2, 7) , (2, 11) , (2, 19) , (2, 23) , (2, 31) , (3, 7) , (3, 23) , (3, 31) , (7, 11) , (7, 19) , (11, 23) ,

(n , 31) , (19, 23) , (19, 31) , (23, 31) .

(3) The resultsof Theorem 5.

The fields with class number 1. ( 3, 11) , ( 3, 19) , ( 7, 23) , ( 7,31) , ( 11, 19) .

(4) The resultsof Theorem 7.

(a) The fieldswitbclassnumber 1.

(2, 3, 5) **, ( 2, 3, 7) **, (2, 3, 13) , (2, 3, 23) , (2, 3, 29) , (2, 3, 31) , (2, 5, 7) **, (2, 5, 11) , (2, 5, 19) ,

(2, 5, 23) , (2, 5, 31) , (2, 7, 11) , (2, 7, 13) , (2, 7 19) , (2, 7, 29) , (2, 11, 13) , (2, n , 29) , (2, n , 31) ,

(2, 13, 19) , ( 2, 13, 23) , (2, 13, 31) , (2, 19, 29) , (2, 19, 31) , (2, 23, 29) .

(b) The fieldswithclassnumber 3. (2, 11, 23) , (2, 19, 23) , (2, 29, 31) .

(5) Theresultsof Theorem 8.

The fields with class number 1. ( 2, 3, 11) **, ( 2, 3, 19) , ( 2, 11, 19) .

(6) The resultsof Theorem 9.

(a) The fieldswithclassnumber 1.

(3, 5, 7) , (3, 5, 11) , (3, 5, 13) , (3, 5, 23) , (3, 5, 31) , (3, 7, 11) , (3, 7, 13) , (3, 7, 17) , (3, 7, 23) ,

(3, 7, 29) , (3, n , 13) , (3, n , 17) , (3, 11, 19) , (3,11,29) , (3,13,19) , (3, 13, 31) , (3, 17, 23) , (3, 17, 31) ,

(3, 19, 29) , ( 5, 7, 11) , (5, 7, 13) , (5, 7, 19) , (5, 7, 23) , (5, 7, 31) , (5, 11, 13) , (5, 11, 17) , (5, U , 23) ,

( 5, 13, 17) ***, (5, 13, 19) , (5, 13, 31) , (5, 19, 23) , (5, 23, 31) , (7, 11, 13) , (7, 11, 17) , (7, 11, 31) ,

( 7, 13, 19) , ( 7, 13, 23) , (7, 13, 31) , (7, 17, 23) , (7, 17, 29) , (7, 17, 31) , (7, 19, 29) , (7, 23, 31) , (11,

13, 17) , (n , 13, 19) , (11, 13, 23) , (11, 13, 29) , (11, 13, 31) , (11, 17, 23) , (11, 17, 31) , (n , 19, 29) ,

( 11, 19, 31) , (11, 23, 29) , (13, 19, 29) , (13, 19, 31) , (13, 23, 31) , (17, 23, 29) , (19, 23, 31) .

(b) The fieldswithclassnumber 3.

(3, 29, 31) , ( 5, 13, 23) , (5, 17, 31) , (11, 29, 31) , (13, 29, 31) , (17, 29, 31) , (23, 29, 31) .

(c) The fieldwith classnumber 7, ( 17, 23, 31) .

(d) The fieldwithclassnumber 9, (whoseideal classgroupisof type (3, 3) )

(19, 29, 31) .

(7) The resultsof Theorem 10.

(a) The fieldswithclassnumber 1.

(2, 3, 5, n ) , (2, 3, 7, 13) , (2, 3, 13, 19) , (2, 3, 13, 31) , (2, 5, 7, 19) .

(b) The fieldwith classnumber 3, (2, 5, 19, 23) .

( *, ** and ¨ ≒ Given in [31 , [7] and [1] respectively.)

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theory (proc. Sci. Res. Counc11AtlasSympos., No2, 0xford 1969) , 153- 165.

[ 2] K. lwasawa, A noteonclassnumbersof algebraicnumber fields, Abh. Math. Sem. Univ. Hamburg20

(1956) 257- 258.

[ 3] T. Kubota, UberdenbizyklischenbiquadratischenZahlk6rper, NagoyaMath. J. 10 (1956)65- 85.

On theclassnumbersand the ideal classgroupsof certain algebraicnumber fields 67

[ 4] N. Nakagashi, A noteon/-dassgroupsofcertainalgebraicnumberfields, J、NumberTheory19 (1984)

140- 147.

[ 5] K. 0hta, 0nthey dassgroupsofaGaloisnumberfieldanditssubfields, J. Math. Soc. Japan30 (1978)

763- 770.

[ 6] K. Uchida, lmaginaryabeliannumberfieldswithclassnumberone, T6hokuMath. J. 24 (1972) 487- 499.

[ 7] H. Wada, 0ntheclassnumberandtheunitgroupofcertainalgebraicnumberfields, J. Fac. Sci. Univ.

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[10] A. Yokoyama, 0ntherelativeclassnumberof finitealgebraicnumber fields, J. Math. Soc. Japan19

(1967) 179- 185.