Table Of ContentHelmut Hasse
On the
Class Number
of Abelian
Number Fields
Extended with Tables by
Ken-ichi Yoshino and Mikihito Hirabayashi
On the Class Number of Abelian Number Fields
Helmut Hasse
On the Class Number
of Abelian Number Fields
Extended with Tables by Ken-ichi Yoshino
and Mikihito Hirabayashi
123
HelmutHasse
Hamburg,Germany
Translatedby
MikihitoHirabayashi
KanazawaInstituteofTechnology
Ishikawa,Japan
ThebookisatranslationoftheGermanedition-twochaptersareaddedasreprints.
ISBN978-3-030-01510-7 ISBN978-3-030-01512-1 (eBook)
https://doi.org/10.1007/978-3-030-01512-1
LibraryofCongressControlNumber:2018961405
MathematicsSubjectClassification(2010):11-XX,11Rxx,11R37,11R29,12-XX
PartIIChapter4reprintedwithkindpermissionofKanazawaMedicalUniversityfrom:Ken-ichiYoshino
andMikihito Hirabayashi, OntheRelative Class Numberofthe Imaginary Abelian NumberField I,
MemoirsoftheCollegeofLiberalArts,Vol.9,December,p.5–53,©Kanazawa MedicalUniversity
1981.
PartIIChapter5reprintedwithkindpermissionofKanazawaMedicalUniversityfrom:Ken-ichiYoshino
andMikihitoHirabayashi, OntheRelative ClassNumberoftheImaginaryAbelianNumberFieldII,
MemoirsoftheCollegeofLiberalArts,Vol.10,December,p.33–81,© KanazawaMedicalUniversity
1982.Correctedprinting2015
EnglishtranslationoftheGermanreprintpublishedbySpringer-VerlagBerlin,Heidelberg,1985
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Foreword
JacquesMartinethas sketchedthe developmentof the main ideasin the theoryof
cyclotomicfieldsfromthepublicationofHasse’smonographuptotheheydaysof
Iwasawatheory(forapolishedproofoftheMainConjecture,see,e.g.,thebeautiful
book [10]). The notorious conjectures (Vandiver, Greenberg) are still open, and
progresshasshiftedsomewhatto the non-abelianside.Thisnoteis devotedto the
algorithmicpartofthetheoryofcyclotomicfields.
As Hasse explainedin his preface, he was not content with the state of the art
in the theory of cyclotomic fields: Hilbert had erected a magnificent building of
algebraic number theory, which had been crowned by Takagi’s class field theory.
On the other hand, the computation of the main invariants of number fields, such
asthe unitgroupand the class number,was notfeasibleexceptfor fieldsof small
degreeandverysmalldiscriminant.InHasse’sopinion,itwasunsatisfactorytohave
suchagreattoolastheanalyticclassnumberformulawhilehavingtoadmitthatit
wasprettyuselessforactuallycomputingclassnumbers.
Hasse’sgoalwastoinvestigatetheclassnumberformulafromanarithmeticpoint
−
ofview;evenshowingthatthenumberh providedbytheclassnumberformulais
anintegerisquitenon-trivial.Hethenshowedthattheclassnumberformulamaybe
usedforcomputingtherelativeclassnumbersofallcyclotomicfieldsofconductor
≤100.
Hasse’smonographservedasablueprintforthecorrespondingbookonabelian
extensions of complex quadratic number fields by Curt Meyer [45]. The whole
situationinthiscaseismuchmorecomplicated,andMeyer’sbookisalotharderto
readthanHasse’s. Whatis missing arethe manytablesthatHasse providedin his
bookandinparticularthebeautifuldiagramsofthesubfieldsinvolved.
Class NumberFormulas
The statement that the analytic class number formula for quadratic number fields
was first proved by Dirichlet must be taken with a grain of salt, since Dirichlet
v
vi Foreword
worked with binary quadratic forms. This fact makes t√ranslating his result on the
classnumberformulaforDirichletnumberfieldsQ(i, m) (or,ashewouldhave
said, for binary quadratic forms with complex coefficients and determinant m) a
highly non-trivial task, since his class groups of forms correspond to certain ring
classgroupsinthemodernsense.Thegeneralizationsofhisclassnumberformula
tomoregeneralbiquadraticnumberfieldsprovidedbyEisenstein[15],Bachmann
[5] and Amberg[1] are so difficultto translate into the languageof numberfields
that it is hard to say whether their results are corrector not (Hasse alludesto this
stateofaffairsinhisIntroduction).
Hilbert [27] and Herglotz [26] proved these results in the language of number
fields,andthemostprofoundstudyoftheclass numberformulainmultiquadratic
numberfieldsisthelittle-knownthesisbyVärmon[66],whichcontainsmostofthe
resultsthatwerelaterrediscoveredindependentlybyotherauthors.
Computation ofClass Numbers
AlreadyGausshadcomputedextendedtablesofclassnumbersofbinaryquadratic
forms,butheusedhistheoryofcomposition,amethodthatissuperiortotheclass
numberformulaforcomputationsbyhand.EvenafterDirichlet’sproofoftheclass
numberformulain the quadraticcase, the theoryof quadraticformsremainedthe
favouritecomputationaltool.
Similar techniques were not available when Kummer started investigating the
class groups of cyclotomic fields. He proved that the class number h of the field
K = Q(ζ ) of p-th roots of unity admits a factorization h = h+h−, where h+
p p p p p
is the class numberof the maximalreal subfield K+ = Q(ζ +ζ−1), and where
p p
h− is an integerthat can be computedexplicitly.The relative class numberh− =
p p
+
h(K)/h(K )isnumericallyaccessible,andtheplusclassnumberisessentiallythe
indexofthegroupofcyclotomicunitsinsidethefullunitgroup.
UsingthetechniquesprovidedbyKummer,C.G.Reuschle[55](1812–1875),a
teacherattheGymnasiuminStuttgart,computedgeneratorsoftheprincipalideals
ofsmallnormsincyclotomicnumberfieldsQ(ζ )forallprimevaluesm<100as
m
wellasforseveralcompositevaluesofm ≤ 120.Reuschle’scorrespondencewith
KummerwaspublishedbyFolkertsandNeumann[17].
After the publication of Hasse’s book, Schrutka von Rechtenstamm [61] pub-
lished extensive tables of relative class numbers of cyclotomic fields in 1964.
D.H. Lehmer [36] computed many minus class numbers of cyclotomic fields; his
resultswereextendedconsiderablyin[18].YoshinoandHirabayashi[69]expanded
Hasse’s tables, including the diagramsfor the subfields (see Part II of the present
translation).
+
The first calculations of h in some non-trivial cases were done by van der
m
Linden[39],whousedOdlyzko’sboundsforcomputingh+forallprimesp≤163
p
(insomecases,hehadtoassumeGRH).
Foreword vii
GreatadvancesweremadebyR.Schoof[59](seealso[33]and[22]),whowas
able to determine factors of class numbers of real cyclotomic fields, which very
likelycoincidewiththeactualclassnumbers.
Stéphane Louboutin has written a wealth of articles on the computation of
relative class numbers of CM fields using analytic means and has used these
techniques for classifying abelian (and non-abelian) CM fields with small class
numbers;see,e.g.,[40].
Class numbers of cyclotomic fields showed up in investigations of Catalan’s
equationxm−yn = 1.Itisnotdifficulttoreducethestatementthatthisequation
doesnothaveanynon-trivialsolutionstothecasexp−yq =1,wherepandq are
distinctoddprimenumbers.Itcanbeshown(see[7,46])thatifthisequationhasa
non-trivialsolution,thenp |h−andq |h−.Theconjecturethattheonlynon-trivial
q p
solution of the Catalan equation in natural numbersis 32 −23 = 1 was obtained
by P. Mihailescu using Stickelberger’stheorem (see [6, 60] for expositions of the
proof).
For an overview on the determination of class numbers using the p-adic class
numberformula,seetherecentthesisbyZhang[70].
Parityof Class Numbers
Hasse proved in Theorem 3.45 that the class number of Q(ζ ) is odd if and only
n
if h− is odd. The conjecture that h− is odd if p and q = p−1 are both prime
n p 2
emerged in the work of Davis [13] and Estes [14]. This conjecture was proved if
2isaprimitiverootmoduloq byEstes;seeStevenhagen[63]forageneralization
and other useful references.Gras [21] provedan importantduality result between
groupsoftotallypositiveandprimarycyclotomicunits;adifferentproofbasedon
theMainConjecturewasgivenrecentlybyIchimura[30].Yoshino[68]provedthat
the class numberof Q(ζ ) is even if n (cid:4)≡ 2mod4 is divisible by 4 distinct prime
n
factors.
For the state of the art concerning the parity of plus class numbers, and the
determination of the 2-class group of real abelian fields, see the recent thesis by
Verhoek[67].
Structure ofClass Groups
Already Kummer [34] investigated the structure of some minus class groups of
cyclotomicfieldsbystudyingtheactionoftheGaloisgrouponclassgroups;inthis
way,hewasabletoshowthattheminusclassgroupofQ(ζ ),whichhasorder8,
29
iselementaryabelian.Subsequently,Kummer’smethodswererefinedbyTateyama
[64],HorieandOgura[29]andmanyotherauthors.
viii Foreword
Kummer’s result that p | h+ implies that p | h− was provedalgebraically by
p p
Hecke [24], who proved that the p-rank e− of the minus part of Cl (Q(ζ )) and
p p
thecorrespondingranke+ofthepluspartsatisfye− ≥e+.Generalizingresul√tsby
Arnold√Scholz[57],whocomparedthe3-ranksoftheidealclassgroupsofQ( m)
andQ( −3m),Leopoldt[38]obtainedstrongboundsbetweenindividualpiecesof
theplusandtheminusclassgroupsofcyclotomicfields.
Class Number1 Problems
Kummerconjecturedin1851thattheminusclassnumberh− ofQ(ζ ),wherepis
p p
prime,isasymptoticallyequalto
p+3
G(p)= p 4 .
p−3 p−1
2 2 π 2
AnkenyandChowlacouldprovethat
−
h
log p =o(logp),
G(p)
whichimpliesthath− =1foronlyfinitelymanyprimesnumbersp.Granville[20]
p
showedthatKummer’sconjecturesarenotcompatiblewithconjecturesinanalytic
numbertheorythatarebelievedtobetrue.
Masley [41] (see also [42]) determined all cyclotomic fields Q(ζ ) with class
m
number 1 and found that all of them satisfy m ≤ 84. A useful result in this
connectionwasprovedbyHorie[28]:ifK ⊆Larecomplexabeliannumberfields,
thenh− |4h−;thiswasgeneralizedtoarbitraryCMfieldsbyOkazaki[52].
K L
Inthisbook,HassediscussesWeber’sresultthattheclassnumbersofthefields
L of2n+2-throotsofunityarealloddinSect.3.16.HarveyCohn[12]pointedout
n
thatthemaximalreal√subfieldsKn ofLn(cid:2)seem√tohaveclassnumber1.Thisiseasy
toproveforK =Q( 2)andK =Q( 2+ 2),anditfollowsfromReuschle’s
1 2
tablesthath(K )=1.CohnprovedthatthefieldsK eitherhaveclassnumber=1
3 n
or≥ 257.He addsthe remark,“We stillhaveobtainednoevidencetodoubt”that
theclassnumbersofthefieldsK arealltrivial.
n
Fukuda and Komatsu [19] improved Cohn’s result and showed that the class
numbersh(K ) are notdivisible by any prime number< 109. Bauer and van der
n
Lindenshowedthath(K ) = h(K ) = 1;Miller [47,48]provedthath(K ) = 1
4 5 6
and,assumingGRH, thath(K ) = 1.Inaddition,heconjecturesthatallsubfields
7
K ofthecyclotomicp-extensionsforanyprimepandanynhaveclassnumber1.
p,n
Buhler,PomeranceandRobertson[8]haveshownthattheCohen–Lenstraheuristics
predict that h+(pn) = h+(p) for almost all primes p and all integers n, where
h+(pn)istheclassnumberofthemaximalrealsubfieldofthefieldofpn-throots
Foreword ix
of unity, and they remark, “It is possible that there are no exceptional primes” p
atall.
HilbertClass Fields
The fields Q(ζp) with p ≤ 19 have class number 1,√and L = Q(ζ23) has class
number3comingfromthequadraticsubfieldK =Q( −23).Inparticular,weget
the Hilbertclass fields of L and K by adjoininga rootof the polynomialf(x) =
x3−x+1withdiscriminant−23.
In a similar way, we can constructunramified cubic extensionsfor many other
cyclotomic fields. The smallest example of a quadratic number field with class
number5wastreatedbyHasse[23].Nowadays,suchcalculationscanbeperformed
routinelyusingthemethodsdescribedinHenriCohen’sbook[11].
A few other known examples of unramified abelian extensions of cyclotomic
fieldsaregivenbythefollowingtable.
(cid:4) h f |discF|
47 5 x5−2x4+2x3−x2+1 472
79 5 x5−2x4+3x2−2x+1 792
71 7 x7−2x6+2x5+x3−3x2+x−1 713
29 8 x8−4x7+8x6−6x5+2x4+6x3−3x2+x+3 296
31 9 x9−x7−2x6+3x5+x4+2x3−x2+x−3 316
The field F of degree 17 whose compositum with K = Q(ζ ) is the Hilbert
64
classfieldofK wascomputedbyNoamElkies[16]:F isgeneratedbyarootofthe
polynomial
f(x)=x17−2x16+8x13+16x12−16x11+64x9−32x8
−80x7+32x6+40x5+80x4+16x3−128x2−2x+68
andhasdiscriminant|discF|=279.
Familiesofunramifiedextensionsofcyclotomicnumberfieldswereconstructed
byArnoldScholz([58];seealso[37]).AninvestigationofMetsänkylä’sresults[44]
onprimefactorsof theminusclassnumberofcyclotomicfieldsfroma classfield
theoreticalpointofviewmightbearewardingproject.
Themaininvestigationofunramifiedabelianextensionsofcyclotomicfieldswas
doneinconnectionwithp-classgroupsofQ(ζ ).Kummerhadalreadyshownthat
p
thesep-classgroupsaregovernedbythedivisibilityofBernoullinumbersbyp,and
someof hisresultsmaybeinterpretedasan explicitconstructionofp-class fields
ofQ(ζ ).
p
ThepreciseconnectionbetweenthedivisibilityofBernoullinumbersandcertain
piecesofthep-classgroupofQ(ζ )wasinvestigatedbyPollaczek[53],Morishima
p
x Foreword
[50,51]andVandiver[65],buttheclearestexpositionandthemostcompleteresults
concerningthiscorrespondenceweregivenbyHerbrand[25].ForK = Q(ζ ),set
p
G=Gal(K/Q),A=Cl(K)/Cl(K)p andσ (ζ)=ζa.Let
a
A ={c∈A:σ (c)=cai forallσ ∈G}.
i a a
ThenA isthepartofAfixedbytheGaloisgroup,andwehave
0
A=A0⊕A1⊕···⊕Ap−2.
It is easy to see that A = A = 0. Herbrand proved that if A (cid:4)= 0 for some
0 1 i
oddindexi, then p | Bp−i, which refinesKummer’sresultthatif p | h−p, then p
divides some Bernoulli number. Since the plus part of the class group is the sum
of the A with even index, Vandiver’s conjecture states that A = 0 for all even
k k
0 ≤ k ≤ p − 3. Herbrand proved that if p | Bp−i, then Ai (cid:4)= 0 if Vandiver’s
conjectureholds,andRibet[56](seealsoMazur’sbeautifulsurvey[43])succeeded
ineliminatingVandiver’sconjectureusingmodularforms.
Inrecentyears,itwasdiscoveredhowtousealgebraicK-theorytoproveresults
abouttheAi.Kurihara[35]wasabletoshowthatAp−3 =0,andSoulé[62]showed
thatAp−n =0foroddvaluesofnsatisfyinglogp>n224n4.
The triviality of certain pieces of A is also related to a conjecture of Ankeny,
√
Artin and Chowla [2–4], according to which the fundamentalunit ε = t +u p
forprimesp ≡ 1mod4satisfiesp (cid:2) u.Kiselev[31],Carlitz[9]andMordell[49]
proved independentlythat this is equivalentto p (cid:2) B(p−1)/2. This conjecture was
verifiedforallprimesp <2·1011in[54].
Forasurveyonthefascinatingconnectionsbetweenvaluesofzetafunctionsand
algebraicK-theory,seeKolster[32].
Jagstzell,Germany FranzLemmermeyer
References
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Quadratwurzelnzusammensetzen.Diss.Zürich1897
2. N.C.Ankeny,E.Artin,S.Chowla,Theclass-numberofrealquadraticnumber
fields.Ann.Math.56,479–493(1952)
3. N.C. Ankeny,S. Chowla,A noteon the class numberofrealquadraticfields.
ActaArith.6,145–147(1960)
4. N.C.Ankeny,S.Chowla,Afurthernoteontheclassnumberofrealquadratic
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