On the class numbers and the ideal class groups of certain algebraic number fields

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Title algebraic 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., Gifu Univ.)

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。

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On the class numbers and the ideal elass groups of certain algebraic number fields

57

K iichiro OH T A

§1. 1ntroduction.

ln genera1, 1et たbe an algebraicnumber field of finitedegree. T hen, the absolute ideal class group andtheclassntlmber of カ will bedenotedby G and 瓦 respectively. Next, for any prime number ♪the♪-Sylow subgroup of (心 will becalled the夕Xdassgroup of ルand wm bedenoted by ら (夕) , whoseorder will bedenotedby 瓦(夕) . M oreover, let瓦 beaGaloisextensionof finite degree over ん. T hen, the subgroup of all ideal classes of G ( 瓦) which are ambigous with respect to K 陳 w ill be called the am bigous y class group of 尺 w ith respect to A・ and w m be denoted by y1衣:♪) .

N ow, let 尺 be a Galois extension of degree 刀 over ヵ and let ヵ be a prime number prime to が. T hen, the following two facts are fundamental in this paper. N amely ;

(1) The restrid ion of thenorm map NIい : G (♪) → G (♪) to y11(♪) is an isomorphism from 凡 (夕) onto G (夕) . ( d . Yokoyama [10D

(2) lfj : G (夕) →G ひ) isthehomomorphism of y dassgroupof んtothat of 尺 inducedby extension of ideals, then w e have j ( G ( ♪) ) Ξ G ( ♪) . ( cf. N akagoshi [ 4] )

Since j ( G ( 夕) ) ⊆ノ14( が) we have 几 ( 瓦) = j ( G ( ♪) ) clearly. T hus, considering G ( ♪) as a subgroup of G ( ♪) , we may put G ( 瓦) = y14( 瓦) in our case・

ln this paper, first, we shan deal with the case where 尺 is a relatively abelian extension of degree 戸 over ヵ whose Galois group G ( K yk) isof type ( 口 , … , /) バ is a prime number. F or the special case where 辨 = 2 we have proved in [ 5] the follow ing theorem . N amely ;

T HEOREM A . L eほ be 回向 eb屈 c m ・mbey 徊 ld of 鋤 ite degyee, 回 d let K be a y血白幽 abdiall a tetlsioR of degyee p ol ey k uXhose Gd ois gγo呻 G ( K Zk) 包 oj̀ 節 e ( 口 ) j is a y ime H s & れ £ d F 1, 瓦 , … , F tu be tk 但 o佃 y inteymd 斌 ej d ds be励 ea k n d K 、町い s α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 (♪) .

U sing what we mentioned above, we may replace ノ11( ♪) and y1ぶ :蕉) ( f= 1, 2,… , /+ 1) by G (夕) and Gべ夕) ( f= 1, 2,…丿 + 1) respedively andrとwriteaboveformula asfollowing ; ( cf.

Nakagoshi 圃 )

G (♪) / G (夕) = G X♪) / G (が) x C乱(夕) / G (夕) ×… x G 。(♪) / G (夕) .

T he first aim of this paper is to extend above theorem to the general case where 琲 ≧2.

Namely, in §2 weshall provethe following theorem.

T HEOREM X、L et k be 回向 eb面 c lmmbey μdd of 畑治 degyee, 回 d ld K be a 畑 a面幽

abdiaR a te肴sio肴of degyeer olX ey k uJhoseGdoisgγ o呻 G ( K 陳 ) zs が㈲ g ( 1バ, …バ ) バ后

Dep. of M ath., F ac. of Gene. E duc. , Gifu U niv

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T HEOREM 2. L d K be n abd n exte, lsion 可 degyee ? ʻ ow r Q uJhose Galois gyo呻 G (K ZQ) is of 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y i = 1, 2,…, /, 法四回加回

民 = 既 ×瓦 ×… ×μ ( diyed 加o血 d ) .

Moyeow,・, が lhe山 ss u mbey廠 of K isodd, tha da oh g theoyde7 0j̀ Bj )y b沃仁 1, 2, …。 )

we have

hχコbl硲 ‥砿

N ow, it follows immediately from above theorem that if we know the class number 垢 of 瓦 is odd, then w e are able to determine the class number and the idei l class group of 瓦 explicitely by considering only the qUadratic subfields of 瓦 H ence, in §3 we shall investigate to construct the num ber field 瓦 w hose classl num er is odd foT 枇 = 2, 3 and 4.

Finally, in§4 weshall givesomenumerical eχmplesandlistthereal number fieldswithodd d ass number in a table for 附 = 2, 3 and 4.

§2. Proof of Theorem l and 2.

First, we recall the following definition and theorem discussed in [ 51, which are important tools to prove T heorem 1・

DEFIN汀ONべd. [5] ) Leは ben α胎わ咄 cu mbeyj eld oj̀ ji咄 edegyee皿 d ld K わ ea

G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 0w y 12 1f tk GdOiS舒O呻 G = G (瓦μ ) sd 球es 服到 Io面昭 co姐伍0 ;

(A) G kαs α肴oymd s油訂o砂 N of oyぬy n alld 筧s油乎o卯s H1, H2, …, Hn可 sαmeoydey m sw k tk t we 随従

GこNH1こ NH2こ …こNHnα 祖 ∩ 瓦 = 佃 } 力r f≠j, 抑陥犯側edmoteby E 匝eM戒t deme戒

Of G.

T HEOREM B . ( cf. [ 5D Leは be n α胎 b面 c m4m bey j μd of j 油 e d昭yee 回心 d K be n (A) -at匹sio肴 of degγeemR o叱y k Ld F, L1,L2, …・玩 be服 s油βdds of K con刈)回 di昭托s佃 c面 dy to tke s油 gyo呻 s N, H 1, 瓦 , …, 瓦 Of 加 GαlOiSF O呻 G ( K Z& ) bjy 伍e Gd ois tk o巧.

lミf p is a l)γlme 侃m

α鈴 ime m4琲bey. Ld F 1, F2, …, F t, 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e mtmbey 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) x Ch(j) ) /G (φ) ×…x C衣j) ) / G (j ) . Moyeopey, u)e k w tk dαss m tmbeγ ?・eld ion αs 知 110面 14g ;

x

臨(友) = ( 皿恥心)) ) /( /4(yヴー1。

N eχt, let 尺 be an abelian eχtension of degree 2・ over the rational number field Q whose沁 1

Galois group G ( 尺 / Q) is of type ( 2, 2, … , 2) . T . K リbota [31, H . W ada [7] and H . Cohn [ 1]

studied the class number of 瓦 by using its unit group and calculated several numerical examples for the case where 辨 = 2 and 3. 0 n the other hand K . U chida [ 6] investigated the case where 瓦 is imaginary and obtained almost all of such 瓦 whoseclass number is one.

T he second aim of this paper isto study the class number and the ideal classgroup of such number field 瓦 as described above without using its unit group. N amely, as an application of T heorem l we shall also prove in §2 the following theorem.

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On the class numbers and the ideal class groups of certain algebraic number fields 59

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

H ere, fronl sanle reason as before w e m ay 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 T heorem A for the case where 辨 = 2, we may let 撰 > 2 and assume that our assertion is true for 附 - 1.

L et F be any one of 瓦 ( 1≦ f≦ O and fix it at a time, W edenotetheGaloisgroup G ( K ZF ) by χ T hen Ⅳ is an abelian group of order 戸一1 and of type ( 1バ , ‥・バ ) . N ext, 1et L 1, L2, … ,

£ t, where X7 ( 政 - 1) / ( Z- 1) , betheintermediatefieldsbetween んand 尺 such that wehave [ 乙

丿刎 = /゛ l for j= 1, 2,…バ. Sinceonly s= ( /ʼ-1- 1) /( X- 1) £ fwehaveF ⊂乙⊂瓦 it follows

immediately that there exist exact g= 乙- s= M - 1 £ j for w hich w e have F n 乙 = ん. F or convenience w e m ay denote such g 句 by L1, L 2, … , L zj respectiveny. M oreover, we denote the Galois groups G ( K Zk) and G ( K yL5) (j = 1, 2, … , め by G and 罵 ( j = 1, 2, … , u) respectively.

Since we have F n 乙 = ん for y= 1, 2,… , g and Lj、L jl = 尺 for j 1≠ふ it fonows immediately that wehave Gこ N111= N瓦 = … = jV7亀 and 瓦丿瓦 , = { ε} forjl ≠ふ wherewedenoteby E theunit element of G. H ence, considering the order oI N and 瓦 (j = 1, 2, … , め resped ively, it follow s easily that 尺 is an ( A ) -extension of degree μ = & over ん. A pplying T heorem B to our case we have

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

N ext, 1et F ʼ be also any one of 瓦 ( 1≦ f≦ O and we assume F ʼ≠F . T hen, since 尺 is an abelian eχtension of degree 戸一2 0ver F F ʼ whose Galois group is of type ( 口 , … j ) , it is easily seen that there exist only z7= ( /ʼ-2- 1) /( Z- 1) 乙。佃 < j ≦O such that we have FFʼ ⊂ ち . T hus, for exact s- zノ= 戸一2 乙 ( 1≦j ≦ 砥) we have F ʼ ⊂乙 and hence C副 ,1) ) ⊆ (気心) ) clearly. l f we assume F = 瓦, then, as wemay let F ʼ be any oneof 瓦 with y≠jl we have

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

and hence

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

⊆< G ズダ) ,…, G 刀 )) > /G ( ♪) d early.

0 n the other hand, if we denote s quadratic subfields of 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. Since we have 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 we have from ( 2. 2)

(ふ(♪) …G ,。(夕) G バ:夕) …G か ) /G (夕) = < Q , (夕) , …,CU(少) > /G (夕) dearly. N ow, from ( 2. 1) and aboveformula we obtain

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(direct product) .

G=几G(♪ )

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

As瓦 is any oneof Zquadraticsubfields of 瓦 thisformula impliesour assertion istrue, that is, G (♪) / G (瓦) isdecomposed intothedirect product asfollowing ;

G (瓦)/G (友) = Ch(加 /G (φ) x C以j) ) /G (瓦) ×… x C球j) ) /G (j ) . T hus, 0ur theorem is proved completely.

PROOF of T HEOREM 2. 1n general, let ルbe an algebraic number field of finite d6gree. T hen, it is w ell know n that we have

H ence, to prove our theorem it is sufficient to show that for any odd prime number カ we have

ln addition to T heorem 2 we shall mention to the special case w here w e have 臨 = 1.

N am ely ;

COROLLARY , N o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. T his coronary follows immediately from T heorem 2.

§3. 0n certain number fields with odd class number.

ln this section we shall investigate the concrete case where the class number is calculated explicitely by T heorem 2. N amely, we shall construct the number fields of degree 4, 8 and 16 whose class numbers are odd.

㈲ Thebiquadratic case.

F irst w e shan prove the following theorem .

T HEOREM 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 L d ねこ Q ( , /万 ) be α qu dm tic sttbβdd 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 s o心 ,

PROOF. W e assume our assertion is not true. T hen we have 臨 = 2・・& with , x≧1 and (2, 4 ) = 1. LetL bethesubfieldof theabsoluteclassfieldof 五7suchthat£ ⊂£ and [1 : f ] = 2”. T h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 , wherewedenoteby E theunitelement of G, weknow theexistenceof

normal subgroup 耳 of G such that G ( K ZQ ) ⊂ 亙 ⊂ G and [ H :・ G ( K ZQ ) ] = 2, L et F be the subfield of £ corresponding to H by the Galois theory. T hen, F is Galois over Q because 耳 is norm al in G. M oreover, it is easily verified that F is an unramified extension of degree 4 0ver ゑ A s the group of order 4 1s abelian, it follows immediately that F is contained in the absolute class field of 尨 H ence, 瓦 must be divided by 4. T his is a contradiction clearly.

T HEOREM. 4, L d K こ Q ( √瓦 √R ) be α biq回 dyd c u m bey j d d, 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) is oddバhm weαssu琲eか三3 (mod 4) a j 加三5 (mod 8) . 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,

T o prove this theorem we need the following lemma due to K . lwasaw a [21.

LEM MA 1. ( cf. lw asaw a [ 2] ) Leは be 朋面鋤臨 c lmm bey j d d of 夕咄 e 面 gyee n d ld (direct product) 。 G (が) こCJ j) ) ×(乱(瓦) ×… X G ,(瓦)

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

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K be α11 eχtm sio筒 of degyee l) o叱 y k, wk ye l) is α 餅 ime mtmbey. M oyeo叱 y, 扨e assMme tk t tk ye d sts on砂 os 函 me d屈 soy of k which is mm示砲泌 K . Tka , 仔 hJ s not 函 isible by 飢 th匹厦 is also 筒ot d屈 s伽 e by l) .

PROOF of T HEOREM 4.

(1) The case♪= 2, The prime idea1(2) of Q is totally ramified in 尺 and hencewehave (2)= p2 1n 瓦 wherepis a prime divisor of 尨 M oreover it it easily verified thatpis the only prime divisor of カ which is ramified in 尺 A s 砥 is odd by our assumption, applying L emm a l to our case our assertion follow s im m ediately.

(2) The caseZ・ミ 3 ( mod 4 ) and 加三5 ( mod 8) . Theprimeideal (2) of Q isdecomposed into prime divisors of K as follow ing ;

(2) = p2, 靖 /(汐= (2)2.

Since the ideal (2) remains as a prime in ゐ, it is easily seen that (2) is the only prime divisor of ん which is ramified is 尺 N ow, 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(m o d4). Ldk=Q(√ J) b eα q 皿d m tics 油 μ ddo fK. Th a,ijT回

l sume 臨 is odd, then hs is d o odd.

PROOF. A s we have ( j ) = - 1 by our assumption, the prime ideal (φ) が Q remains as a prime in ん atld it is ramified in 尺 clearly. 0 n the orher hand the prime ideal (2) of Q is ramified in ルbecause we have αミ 3 ( mod 4) by our assumption. M oreover, if ♪( L ( mod 4) , then, the idea1 (2) is unramified in Q ( √j ¯) . lf 夕三3 ( mod 4) , then wehave 砂三1 ( mod 4) and hence the ideal (2) is unramified in Q ( y万 ) . F rom these facts it is easily seen that the prime divisor of (2) in A・isunramified in 瓦 Hence (♪) istheonly primedivisor of んwhich isramifiedin 瓦 N ow , 0ur assertion f0110ws immediately from L emma 1.

T HEOREM ら. L d 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)yime mtmbeys st4ck tk t l) ミ q三3 ( mod 4) . L et k = Q ( 刀海) be the q皿 dm tic su吊 dd of K 、 Then , if we αssume hJ s o服 , tka k is 幽 o odd.

PROOF. Since w e have 知三付) 三 - 9 ミ 1 ( m od 4) from our assumption, the prime ideal (2) of Q is unramified in 瓦 H ence, it is easily seen that all finite primes of ヵ are unramified in 瓦

N ow , w e assunle our assertion is not true. T hen, it followsby the same w ay as in T heorem 3 that there exists an unramified extension of degree 2 0ver 尺 such that it is also Galois over Q. W edenoteitby £ . Let F bethemaximal real subfield of £ . Then, itfollowsimmediately that F isan unramified extension of degree 2 0ver 尨 T hisimpliesthat 瓦 isdivisibleby 2. T his is a contradiction clearly. T hus, our theorem is proved completely.

(b) The octic case.

F irst w e shall prove the following theorem. N amely ;

T HEOREM 7. L et K = Q ( √2¯, √j ¯, √i ) be 皿 od c μd d whose G山 is F o呻 { s 原図 } e ( 2, 2, 2) , wheyel) is d hey eq皿に o - 1 0y 皿 odd l)百me筒m (mod 4) α肩 q is 皿 odd 函 me u mber d胎 ye戒斥om 払 M oyeo斑冶 k y 加ミ 5 0y qミ 5 ( mod 8) . Let k= Q ( ,/¯y, √i ¯) betkebiq皿dmtic誹吊 dd of K . Thm , if 扨eassRmekμs odd, th匹 k iS d O Odd.

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PROOF. F irst 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) . H ence it follows immediately that ideal (2) is decomposed into the prime divisors in 瓦 as following ;

(2)= 畢4, yVk/。畢= (2)2.

1f we denote by p the prime divisor of (2) in ん, then we have 2 = 岬, Λ1/。抑= (2)2 clearly. T hus, it is easily verified that p is the only prime divisor of ゐ which is ramified iil 瓦 A pplying L emma l to our case our assertion follows at once.

N ow , to prove neχt theorems we shall need the following lemma. N amely ;

L EM MA 2、 L d & be 皿 d geb咄 c mtm ber 加は of β池 e degyee 皿 d let K be a Gd ois a tm sioll of 面 g粍e 8 0叱 y k、 L d F be 朋 i戒 eym d id e j d d be励 eeu lこ α11d K sMch tk t

[F : 列 = 4皿d theGdois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 is α函 mH dd of F wk H s mlyα琲浜ed 泌 K, tk 11K is αu αbdiα筒a te肴sioM owy k、

PROOF. A s the Ga101s group G = G ( K Zk) is of order 8, to prove our lemma it is sufficient to show that neither the quaternion group nor the dihedral group of order 8 1s isom orphic to G.

Since the GaIois group G ( F μ ) is of type ( 2, 2) by our assumption, there exist 3 proper intermediate fields between ヵ and F . W e denote them by £ l, £ 2 a姐 £ 3 respectively. T hen, it is easily verified from our assumption that p is not decomposed into the product of different prime ideaIs in any L i (しi = 1, 2, 3) .

N ow , we shall consider the decomposition of p into the prime divisors of 瓦 T hen, from our assumption wehaveeither p = 雫2, yVk/h雫= p4 0r p二(雫1叩2) 2j V臨畢f= p2 clearly. lf we have the first case, then w e denote the inertial field of 畢 by 7こ T hen, it fo110ws immediately that the Galois group G ( T ノk) is a cyclic group of order 4, that is, 7̀ is a cyclic extension of degree 4 0ver 瓦。 0n the other hand if wehavethelatter case, then w edenotethedecomposition field of 畢 l by 乙 . T hen, it is easily verified that 乙 is an extension of degree 2 0ver ん which is different from any L バ ,i = 1, 2, 3) . T his implies that there exist at least 4 proper intermediate fields between ル and 尺 each of which is an extension of degree 2 0ver 尨

N ow , we assume G is isom orphic to eiter the quatem ion group or the dihedral group of order 8. T hen, it f0110ws immediately that there exists no cyclic extension of degree 4 0ver ん and there exist only 3 proper intermediate fields between ん and 尺 each of which is an extension of degree 2 0ver 力. T his is a contradiction clearly. H ence, G must be abelian group。

T HEOREM 8, L d K こ Q ( √2¯, √j T, √i ) be 皿 od c j d d w加 se Gd ois F o砂 is 可幼 e (2, 2, 2) , Wk yej) is dtk y 同心に o - 1 0y 皿 odd j)百観em4mbey sttch tk t j) ミ

ん干Q ( √が, √i ) b elhebiqudmtics14球eld of K. Tha , が回 ass1 4me玩 is

αlSO Odd、

PROOF. VVe assunlle our assertion is not true. T hen, it follows by the same way as in T heorem 3 that there eχists an unramified extension of degree 2 0ver 瓦 which is Galois over Q. W e denoteit by £ .

A s we have G ) = 1 from our assumption, the prime ideal ( g) of Q splits in O ( √D /and we have ( g) = pl 芯 g加 g ㈲ : に 1, 2) is a prime ideal of Q ( √j T). M oreover, the ideal (瓦)

od 4) a j

| = 1. £d

tk 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 1

1̲ 八 y m m χ 1 . 1 1・ . . . . . . . . . ̲̲ - = - - g

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ideal of 尺 such that we have jVkyQ( √が) 0 1= q12, SinceO l is unramifiedj n L , applying L emm a 2 to our case, it follow s im m ediately that £ is an abelian extension of degree 8 0ver Q ( √j ¯) .

0 n the other hand, as we have 匹 3 ( mod 4) from our assumption, we may put (2) = p2 1n

Q ( √y) , where pisaprimeideal inQ ( √j ¯), Moyeowy, astk ided (2) isltllmm涵ddthey 加 Q ( √万) or in Q ( 刀海) according to 9三1 0r 9三3 (mod 4) , it is easily seen that pis

unramified in ん. But, since the ideal (2) is totally ramified in Q ( √2¯, √j ¯) , it f0110ws that the prime divisor of p in ん must be ramified is 尺.

N ext, since the Ga101s group G ( L yQ ( √7F) ) is abelian, it iseasily verified that the inertial field of any prime divisor of p in £ with respect to L ZQ ( √j ¯) is uniquely determined. W e denoto it by 7こ A s the prime divisor of p in 瓦 is unramified in £ , it fonows easily that、T is a proper intermediatefieldbetween ヵ and £ andwehave尺 ≠乙 Asany primeideal of んwhich does not divide p is unramified in £ , it f0110ws at once that T is an unramified eχtension of degree 2 0ver た T his implies that 瓦 is divisible by 2. T his is a contradiction clearly. T hus,

0 u r th eo r em is p r ov ed com p l etel y .

T HEOREM 9. L et K = Q (: √万 , √万 , √ y¯) be 皿 oc励 μd d, tりk ye l) is d服 y eq皿に o - 1

0 y α11 0 d d l) yi m e 肴 m

仁 y j l

乙g

ぐ j

is ramified in Q ( √i ) and ramains as a prime in Q ( √2¯) becausu we have G ) こ - 1 from our assumption. H ence it is easily verified that we have q1= こ 12 1n 瓦 where£ l is a prime

糾観

is oddバ hm

抑 e αssR 衿1e

PROOR W e assurne our assertion is not true. T hen, it follows by the same w ay as in T heorem 3 that there eχists an unramified eχtension of degree 2 0ver 尺 which is Galois over Q. W e denote it by £ .

Since we have G ) = 1 from our assumption, the prime 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 as we have ( £ ) = - 1 from our assumption/ it folloJ s easily that qf ( i= 1, 2) rem ains as a prime in Q ( √j ¯ノ √F) . qf ( j = 1, 2) is ramified in 瓦 dearly and hencewe haVe ql =

/

a12, 篤 /, ( √j ¯) ら 1= q12, wherea l isaprimeideal of 尺 which isunramifiedin£. Now,

applying L emma 2 to our case, it follows immediately that the Galois group 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, the prime ideal ( r) of Q remains as a prime in Q ( √y ) and it splits in ゑ H ence wemay put (y) = rl r2, where tl and r2 are different prime ideals of ヵ which areramified in 瓦

N ext, since the Galoisgroup G ( L yQ ( 巧 ) ) is abelian, it is easily verified that the inertial field of any prime divisor of ( y) in £ with resped to L yQ ( √j ¯) is uniquely determined. W e denote it byT oT hen, it follows easily by the samew ay asin T heorem 8 that T isan unramified extension of degree 2 0ver ゐand hence 瓦 must be divided by 2. T his is a contradiction clearly.

T hus, our theorem is proved completely.

(mod 4) , 肋a u 。x吋加叱に 1 (mod 4) . 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 iS dS0 0d1

= 1 α77j = - 1.

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(C) T he real biquartic case.

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

T HEOREM 10. L d K = Q ( √2¯, √j ¯, √肌 √ F¯) be a yeαl mlm bey j d d of degyee 1 6 whose Gd ois gyo呻 is of 削)e 収 , 2 , 2, 2) , where p, q and r are different odd prime numbers such that we have p三r 三3 ( m od 4) , ・7こ 5 ( m od 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g gg瓦八心必法四尻 £sα励

Odd、

PROOF. W e assunle our assertion is not true. T hen, it follows as before that there exists an unramified extension of degree 2 0ver 瓦 which is Galois over Q. W e denote it by £ .

A s we have Z・三 y三3 ( mod 4) and 9 三5 ( mod 8) from our assumption, applying the quadratic reciprocity 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 we have( D = - l fronl our assumption, 則 remains as a primein Q ( √2¯, √万, √F) . As 、pl isramified in K , it iseasily seen that w e have p 1二雫 12, 裁 /。( √万。芦 ) 雫 1= 貼 2, w here 剔 is a prime ideal of 尺 which is unramified in L . N ow , applying L emma 2 to our case, it follows immediately that the Galois group G ( LZQ ( √示, √F) ) is abelian.

On the other hand, it is easily verified by our assumption that the prime ideal (2) of Q is decomposed into prime divisors in Q ( , /万 , √7 ) as following ;

(2)= q2, 蹟 ( √i, √7) /Q q= (2)2

M oreover, it fonows easily that qsplits in Q ( √が, √y, √ F) because the inertial degree of (2) with respect to Q ( √j ̀, √iy¯, √F) / Q is equal to 2. 0 n the other hand, the ideal (2) is totally ramified in Q ( √2¯, √ F ) , and hence ramification index of (2) w ith respect to 瓦 / Q is at least 4. T hen, the prim e divisor of q in Q ( √j jʼ, √i , √ F¯) m ust be ram ified in 瓦・

N ext, since the Galois group G ( L ZQ ( √y, √ F) y is abelian, it is easily verified that the inertial field of any prime divisor of q in l with resped to L ZQ ( √y, √F ) is uniquely determined. W e denote it by T 、 T hen, it follows easily by the same way as in T heorem 8 that T is an unramified extension of degree 2 0ver Q ( √j ¯, √ 4¯, √ F ) and hence the class number

of Q ( √j ¯, √¯y, √¯ F) mustbedividedby2. But, sinceitiseasilyseenthat all assumption

of T heorem g are satisfied in our case, it follows immediately that theclassnumber of Q ( √j ¯,

√i , √¯ F) isodd. Thisisacontradictionclearly. Thus, our theorem isprovedcompletely.

§4. Examples.

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

y z( √瓦 √4¯) respectivelyinthefollowing exmples.

(1) K 二Q ( √5¯, μ7 ) . W ehaveyz( √副) = k ( μ7) = 1 and yz( √器5¯) = 6. Hencefrom T heorem 3 w e huve 臨 = 3.

(2) 瓦= Q ( 1, √面¯r¯) . W ehaveyz( √i ) = み( √皿 [ ] = 1 and yz( √顎T) = 14. Hencefrom T heorem 3 we have 厦 = 7.

(3) £ = Q ( √T , μ¯ r) . W ehave71- 3 (mod4) 丿 ( √2¯) = y z( J T) = 1 and y z( √R y ) = 3.

H ence from T heorem 4 w e have 臨 = 3.

(4卜 £ = Q ( y7 , √ B¯[ ] . W e have 151 三 3 (mod 4) , 453 三 5 (mod 8) and yz ( √1¯) =

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

(13) 尺= Q ( √3¯, √lヨ F, √Γ Υ ) . Wehave(

=6. H ence from T heorem 8 w e have 臨 = 3.

(jy)=(サ

^{alid yz (}

ム

15) = yz ( √B S ) = 2. H ence from

万一 1a n d y z (√ 5 ¯ ,√ ri)=1fro m

T heorem 3. N ext we have yz ( √3¯) = yz ( T heorem g w e have yzχ= 1.

y z( √mで r¯) = y z( √B3¯= 1. Hencefrom Theorem 4 wehave尻こ1.

(5) 尺= Q ( f, yR ) . W ehave 83三3 ( mod 4) , -83三5 (mod 8) 丿 ( √T) = yz( J 3) = 1 and

y z( √二羽¯) = 3. Hencefrom Theorem 4 wehave臨= 3.

(6) X = Q ( √y, √胚F) . Wehave439 - 3 (mod4) , (響 ) = -1丿 ( √r) = y z( y剪惣) = 1and y z( √認9¯) = 5. Hencefrom Theorem 5 wehave尻= 5.

(7) 尺= Q( j, √2213) . W ehave( 忌 ) = -1j z( √i) = 1, y z( √2酉¯) = 3 and ゐ( √223) = 7 .

H ence from T heorem 5 we have 尻 = 21.

(8) 尺= Q ( √3¯もぷ1お) . W ehave23- 3 (mod4) , み( y弱) = み( √3) = 1 andy z( √二 23¯) =

3. H ence from T heorem 6 w e have 尻 = 3.

(9) 瓦= Q ( √2¯ √5¯, J ¯9) . First we have y z( √15¯, √[9] = 1 from Theorem 3. Next we

have yz ( √亙) = yz ( y盾 ) = 1 四 j yz ( √皿 ) = yz ( √n O¯) = 2. H ence from T heorem 7 we have 尻= 1.

(10) X = Q (i, √2¯, √mi ) . Firstwehavey z( √T, √T皿 = 7 from above(2). Nextwehave y z( √r ) = y z( √2) = 1, y z( √涯2¯) = 2 and y z( √2屁) = 6. HencefromTheorem7wehave瓦=

21.

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

T heorem 3. N ext we have yz ( J 2) = 1 and yz ( J ¯{} ) = yz ( √i¯n) ) = 2. H ence from T heorem 8 we have 尻 = 3.

㈲翼=Q(と√ 2¯ , √ r3). Weh a v e(il) ⊃1,(且)=1a n dノ z(√i¯ , yn)=1fro m[61.Ne xt

(14) j l (mod and yz (

宍=1,(y

-11) = 1, み(

F )=シ χ

-143) = 10. Hencefrom Theorem gwe

J A ?=-1a n d み (√ iy n )=1fro m

㈲瓦 = Q ( j, 、/Ti , 、yr3) . W e have above(12). N ext wehave yz( μ Υ) = yz(

have 臨 = 5.

㈲尺 = Q ( √2¯, √3ʻ̀, √7¯, √口) . W eput X・= 3, ・7= 13 and , ・= 7 1n T heorem 10. T hen we

h a v e(il) =1,(で ih)=(j)=-1a n dゐ (√ 3¯,√ r3)=1fro mTh e o re m3.Ne xtitise a s ilys e e n

that there exists no quadratic subfield of 尺 whose class number is divided by an odd prim e number. H ence from T heorem 10 we have 臨 = 1.

Finally, we shaII list in a table the class numbers of real number fields such 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) The results of Theorem 3.

(a) T he fields with class number 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.Ne xtw eh a v eみ (y 37)=1,み (√ 苓iT )=み (√ 629¯ )=2

from T heorem g we have 尻 = 1.

= 4.

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( 22, 29) , ( 23, 29)

(b) The field with classnumber 3, ( 29, 31) (2) The results of Theorem 4.

T he 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) .

T he fields w ith class number 1. ( 3, 11) , ( 3, 19) , ( 7, 23) , ( 7,31) , ( 11, 19) . (4) The results of T heorem 7.

(a) The fields witb class number 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 fields with class number 3. ( 2, 11, 23) , ( 2, 19, 23) , ( 2, 29, 31) . (5) The results of Theorem 8.

T he fields w ith class number 1. ( 2, 3, 11) * *, ( 2, 3, 19) , ( 2, 11, 19) . (6) The results of Theorem 9.

(a) The fields with class number 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 fields with class number 3.

( 3, 29, 31) , ( 5, 13, 23) , ( 5, 17, 31) , ( 11, 29, 31) , ( 13, 29, 31) , ( 17, 29, 31) , ( 23, 29, 31) . (c) The field with class number 7, ( 17, 23, 31) .

(d) The field with class number 9, ( whose ideal class group is of type ( 3, 3) ) ( 19, 29, 31) .

(a) The fields with class number 1.

( 2, 3, 5, n ) , ( 2, 3, 7, 13) , ( 2, 3, 13, 19) , ( 2, 3, 13, 31) , ( 2, 5, 7, 19) . (b) The field with class number 3, ( 2, 5, 19, 23) . ( *, ** and ¨ ≒ Given in [31 , [7] and [1] respectively.)

R eferences

[ 1] H. Cohn, A numerical study of unitsin compositereal quarticand octic fields, Computersin number theory ( proc. Sci. Res. Counc11AtlasSympos., No 2, 0 xford 1969) , 153- 165.

[ 2] K. lwasawa, A noteon classnumbersof algebraicnumber fields, Abh. Math. Sem. Univ. Hamburg 20 ( 1956) 257- 258.

[ 3] T. Kubota, Uber denbizyklischenbiquadratischenZahlk6rper, Nagoya Math. J. 10 ( 1956) 65- 85.

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[ 4] N. Nakagashi, A noteon /-dassgroupsof certainalgebraicnumber fields, J、Number Theory 19 (1984)

140- 147.

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

763- 770.

[ 6] K. Uchida, lmaginary abeliannumber fieldswithclassnumber one, T6hokuMath. J. 24 ( 1972) 487- 499.

[ 7] H. W ada, 0n theclassnumber andtheunitgroupof certainalgebraicnumber fields, J. Fac. Sci. Univ.

T okyo Sec 1 13 ( 1966) 201- 209.

[ 8] H. W ada, A Tableofldeal classNumbersof Real OuadraticFields(inJapanese) , LectureNoteinMath.

10( 1981) , Sophia Univ., T okyo.

[ 9] H. W ada and M. Saito, A Table of ldea1ClassGroups of lmaginary OuadraticFields (in Japanese) ,

L ecture N ote in M ath. 28 ( 1988) , Sophia U niv. , T okyo.

[10] A. Yokoyama, 0n therelativeclassnumber of finitealgebraicnumber fields, J. Math. Soc. Japan 19 ( 1967) 179- 185.