Imaginary quadratic fields with isomorphic abelian Galois groups

24 Pages·2013·0.68 MB·English

THE OPEN BOOK SERIES 1
ANTS X
Proceedings of the Tenth
Algorithmic Number Theory Symposium
Imaginary quadratic fields with
isomorphic abelian Galois groups
AthanasiosAngelakis and PeterStevenhagen
msp

THEOPENBOOKSERIES 1 (2013)
TenthAlgorithmicNumberTheorySymposium msp
dx.doi.org/10.2140/obs.2013.1.21
Imaginary quadratic fields with
isomorphic abelian Galois groups
AthanasiosAngelakis and PeterStevenhagen
In1976,Onabediscoveredthat,incontrasttotheNeukirch-Uchidaresultsthat
wereprovedaroundthesametime,anumberfieldK isnotcompletelycharac-
terizedbyitsabsoluteabelianGaloisgroupA . Thefirstexamplesofnoniso-
K
morphicK havingisomorphicA wereobtainedonthebasisofaclassification
K
byKubotaofideleclasscharactergroupsintermsoftheirinfinitefamiliesof
Ulminvariants,anddidnotyieldadescriptionofA . Inthispaper,weprovide
K
adirect“computation”oftheprofinitegroupA forimaginaryquadraticK,and
K
useittoobtainmanydifferentK thatallhavethesameminimalabsoluteabelian
Galoisgroup.
1. Introduction
TheabsoluteGaloisgroupG ofanumberfieldK isalargeprofinitegroupthat
K
we cannot currently describe in very precise terms. This makes it impossible to
answerfundamentalquestionson G ,suchastheinverseGaloisproblemover K.
K
Still,Neukirch [7] provedthatnormalnumberfields arecompletelycharacterized
by their absolute Galois groups: If G and G are isomorphic as topological
K1 K2
groups, then K and K are isomorphic number fields. The result was refined
1 2
by Ikeda, Iwasawa, and Uchida ([8], [9, Chapter XII, §2]), who disposed of the
restriction to normal number fields, and showed that every topological isomor-
phism GK1(cid:0)(cid:24)! GK2 is actually induced by an inner automorphism of G(cid:81). The
samestatementsholdifallabsoluteGaloisgroupsarereplacedbytheirmaximal
prosolvablequotients.
ItwasdiscoveredbyOnabe[10]thatthesituationchangesifonemovesafurther
stepdownfromG ,toitsmaximalabelianquotientA DG =ŒG ;G (cid:141),which
K K K K K
istheGaloisgroupA DGal.Kab=K/ofthemaximalabelianextensionKab ofK.
K
MSC2010: primary11R37;secondary20K35.
Keywords: absoluteGaloisgroup,classfieldtheory,groupextensions.
21
22 ATHANASIOSANGELAKISANDPETERSTEVENHAGEN
EventhoughtheHilbertproblemofexplicitlygeneratingKab forgeneralnumber
fieldsK isstillopenaftermorethanacentury,thegroup A canbedescribedby
K
classfieldtheory,asaquotientoftheideleclassgroupofK.
Kubota[5]studiedthegroupX ofcontinuouscharactersonA ,andexpressed
K K
thestructureofthep-primarypartsofthiscountableabeliantorsiongroupinterms
ofaninfinitenumberofso-calledUlminvariants. IthadbeenshownbyKaplansky
[4,Theorem14]thatsuchinvariantsdeterminetheisomorphismtypeofacount-
able reduced abelian torsion group. Onabe computed the Ulm invariants of X
K
explicitlyfor anumber ofsmall imaginaryquadratic fieldsK, andconcluded from
0
thisthatthereexistnonisomorphicimaginaryquadraticfieldsK andK forwhich
theabsoluteabelianGaloisgroupsAK andAK0 areisomorphicasprofinitegroups.
0
ThismayevenhappenincasepswhereK andKp havedifferentclassnumbers,but
theexplicitexampleK D(cid:81). (cid:0)2/,K0D(cid:81). (cid:0)5/ofthisthatoccursinOnabe’s
maintheorem [10,Theorem 2]isincorrect. This isbecausethe valueofthe finite
Ulminvariantsin[5,Theorem4]isincorrectfortheprime2incasethegrounpdfield
isaspecialnumberfieldinpthesenseofourLemma3.2. Asithappens,(cid:81). (cid:0)5/
andtheexceptionalfield(cid:81). (cid:0)2/dohavedifferentUlminvariantsat2. Thenature
ofKubota’s errorissimilarto anerrorin Grunwald’stheoremthat wascorrected
byatheoremofWangoccurringinKubota’spaper[5,Theorem1]. Itisrelatedto
thenoncyclicnatureofthe2-powercyclotomicextension(cid:81)(cid:26)(cid:81).(cid:16)21/.
Inthispaper,weobtainOnabe’scorrectedresultsbyadirectclassfieldtheoretic
approachthatcompletelyavoidsKubota’sdualizationandthepmachineryofUlm
invariants. WeshowthattheimaginaryquadraticfieldsK ¤(cid:81). (cid:0)2/thataresaid
tobeof‘typeA’in[10]sharea minimal absoluteabelianGaloisgroupthatcanbe
describedcompletelyexplicitlyas
A D(cid:90)y2(cid:2) Y (cid:90)=n(cid:90):
K
n(cid:21)1
Thenumericaldatathatwepresentsuggestthatthesefieldsareinfactverycommon
amongimaginaryquadraticfields: Morethan97%ofthe2356fieldsofoddprime
class number h Dp <100 are of this nature. We believe (Conjecture 7.1) that
K
there are actually infinitely many K for which A is the minimal group above.
K
Ourbeliefissupportedbycertainreasonableassumptionsontheaveragesplitting
behavior of exact sequences of abelian groups, and these assumptions are tested
numericallyinthefinalsectionofthepaper.
2. Galoisgroupsas(cid:90)y-modules
TheprofiniteabelianGaloisgroupsthatwestudyinthispapernaturallycomewith
atopologyforwhichtheidentityhasabasisofopenneighborhoodsthatareopen
subgroupsoffiniteindex. Thisimpliesthattheyarenotsimply(cid:90)-modules,butthat
IMAGINARYQUADRATICFIELDSWITHISOMORPHICABELIANGALOISGROUPS 23
theexponentiationinthesegroupswithordinaryintegersextendstoexponentiation
with elements of the profinite completion (cid:90)y Dlim (cid:90)=n(cid:90) of (cid:90). By the Chinese
(cid:0)
remaindertheorem, wehaveadecomposition ofthenprofinite ring(cid:90)y DQ (cid:90) into
p p
a product of rings of p-adic integers, with the index p ranging over all primes.
As(cid:90)y-modules,ourGaloisgroupsdecomposecorrespondinglyasaproductofpro-
p-groups.
It is instructive to look first at the (cid:90)y-module structure of the absolute abelian
Galois group A(cid:81) of (cid:81), which we know very explicitly by the Kronecker-Weber
theorem. Thistheoremstatesthat(cid:81)ab isthemaximalcyclotomicextensionof(cid:81),
andthatanelement(cid:27) 2A(cid:81) actsontherootsofunitythatgenerate(cid:81)ab byexponen-
tiation. Moreprecisely,wehave(cid:27).(cid:16)/D(cid:16)u forallrootsofunity,withuauniquely
defined element in the unit group (cid:90)y(cid:3) of the ring (cid:90)y. This yields the well-known
isomorphismA(cid:81)DGal.(cid:81)ab=(cid:81)/Š(cid:90)y(cid:3)DQp(cid:90)p(cid:3).
For odd p, the group (cid:90)(cid:3) consists of a finite torsion subgroup T of .p(cid:0)1/-st
p p
rootsofunity,andwehaveanisomorphism
(cid:90)(cid:3)DT (cid:2).1Cp(cid:90) /ŠT (cid:2)(cid:90)
p p p p p
because 1Cp(cid:90) is a free (cid:90) -module generated by 1Cp. For p D2 the same
p p
is true with T Df˙1g and 1C4(cid:90) the free (cid:90) -module generated by 1C4D5.
2 2 2
Takingtheproductoverallp,weobtain
A(cid:81)ŠT(cid:81)(cid:2)(cid:90)y; (1)
withT(cid:81)DQpTp theproductofthetorsionsubgroupsTp (cid:26)(cid:81)p(cid:3) ofthemultiplica-
tivegroupsofthecompletions(cid:81)p of(cid:81). Morecanonically,T(cid:81) istheclosureofthe
torsionsubgroupofA(cid:81)DGal.(cid:81)ab=(cid:81)/,andA(cid:81)=T(cid:81) isafree(cid:90)y-moduleofrank1.
TheinvariantfieldofT(cid:81) inside(cid:81)ab istheunique(cid:90)y-extensionof(cid:81).
EventhoughitlooksatfirstsightasiftheisomorphismtypeofT(cid:81) dependson
thepropertiesofprimenumbers,oneshouldrealizethatinaninfiniteproductof
finitecyclicgroups, theChineseremainder theoremallowsustorearrange factors
inmanydifferentways. Onehasforinstanceanoncanonicalisomorphism
Y Y
T(cid:81)D Tp Š (cid:90)=n(cid:90); (2)
p n(cid:21)1
as both of these products, when written as a countable product of cyclic groups
of prime power order, have an infinite number of factors (cid:90)=`k(cid:90) for each prime
power `k. Note that, for the product Q T of cyclic groups of order p(cid:0)1 (for
p p
p¤2),thisstatementisnotcompletelytrivial: Itfollowsfromtheexistence,bythe
well-known theorem of Dirichlet, ofinfinitely many primesp that are congruent
to1 mod`k,butnotto1 mod`kC1.
24 ATHANASIOSANGELAKISANDPETERSTEVENHAGEN
Now suppose that K is an arbitrary number field, with ring of integers O. By
classfieldtheory,A isthequotientoftheideleclassgroupC D(cid:0)Q0 K(cid:3)(cid:1)=K(cid:3)
K K p(cid:20)1 p
ofK bytheconnectedcomponentoftheidentity. Inthecaseofimaginaryquadratic
fieldsK,thisconnectedcomponentisthesubgroupK(cid:3) D(cid:67)(cid:3)(cid:26)C comingfrom
1 K
the unique infinite prime of K, and in this case the Artin isomorphism for the
absoluteabelianGaloisgroupA ofK reads
K
A DKy(cid:3)=K(cid:3)D(cid:16)Y0K(cid:3)(cid:17)=K(cid:3): (3)
K p
p
Here Ky(cid:3) DQ0 K(cid:3) is the group of finite ideles of K, that is, the restricted direct
p p
(cid:3)
productofthegroupsK atthefiniteprimespofK,takenwithrespecttotheunit
p
groupsO(cid:3) ofthelocalringsofintegers. Forthepurposesofthispaper,whichtries
p
todescribeA asaprofiniteabeliangroup,itisconvenienttotreattheisomorphism
K
forA in(3)asanidentity—aswehavewrittenitdown.
K
Theexpression(3)issomewhatmoreinvolvedthanthecorrespondingidentity
A(cid:81) D(cid:90)y(cid:3) for the rational number field, but we will show in Lemma 3.2 that the
inertialpart ofA ,thatis, thesubgroupU (cid:26)A generatedbyallinertia groups
K K K
O(cid:3)(cid:26)C ,admitsadescriptionverysimilarto(1).
p K
DenotebyOyDQ O theprofinitecompletionoftheringofintegersOofK. In
p p
thecasethatK isimaginaryquadratic,theinertialpartofA takestheform
K
U D(cid:16)YO(cid:3)(cid:17)=O(cid:3)DOy(cid:3)=(cid:22) ; (4)
K p K
p
sincetheunitgroupO(cid:3) ofOisthenequaltothegroup(cid:22) ofrootsofunityinK.
K
Apart from the quadratic fields of discriminant (cid:0)3 and (cid:0)4, which have 6 and 4
rootsofunity,respectively,wealwayshave(cid:22) Df˙1g,and(4)canbeviewedas
K
theanalogueforK ofthegroup(cid:90)y(cid:3)DA(cid:81).
In the next section, we determine the structure of the group Oy(cid:3)=(cid:22) . As the
K
approach works for any number field, we will not assume that K is imaginary
quadraticuntiltheveryendofthatsection.
3. Structureoftheinertialpart
LetK beanynumberfield,andOyDQ O theprofinitecompletionofitsringof
p p
integers. DenotebyT (cid:26)O(cid:3) thesubgroupoflocalrootsofunityinK(cid:3),andput
p p p
T DYT (cid:26)YO(cid:3)DOy(cid:3): (5)
K p p
p p
Theanalogueof(1)forK isthefollowing.
IMAGINARYQUADRATICFIELDSWITHISOMORPHICABELIANGALOISGROUPS 25
Lemma3.1. Theclosureofthetorsionsubgroupof Oy(cid:3) isequaltoT ,andOy(cid:3)=T
K K
isafree(cid:90)y-moduleofrankŒK W(cid:81)(cid:141). Lesscanonically,wehaveanisomorphism
Oy(cid:3)ŠT (cid:2)(cid:90)yŒKW(cid:81)(cid:141):
K
Proof. As the finite torsion subgroup T (cid:26)O(cid:3) is closed in O(cid:3), the first statement
p p p
followsfromthedefinitionoftheproducttopologyonOy(cid:3)DQ O(cid:3).
p p
ReductionmodulopinthelocalunitgroupO(cid:3) givesrisetoanexactsequence
p
1(cid:0)!1Cp(cid:0)!O(cid:3)(cid:0)!k(cid:3)(cid:0)!1
p p
(cid:3)
that canbe splitby mapping theelements ofthe unit groupk of theresidue class
p
field to their Teichmüller representatives in O(cid:3). These form the cyclic group of
p
order #k(cid:3) D Np(cid:0)1 in T consisting of the elements of order coprime to p D
p p
char.k /. The kernel of reduction 1Cp is by [3, one-unit theorem, p. 231] a
p
finitelygenerated(cid:90) -moduleoffreerankŒK W(cid:81) (cid:141)havingafinitetorsiongroup
p p p
consistingofrootsofunityinT ofp-powerorder. Combiningthesefacts,wefind
p
that O(cid:3)=T is free over (cid:90) of rank ŒK W(cid:81) (cid:141) or, less canonically, that we have a
p p p p p
localisomorphism
O(cid:3)ŠT (cid:2)(cid:90)ŒKpW(cid:81)p(cid:141)
p p p
for each prime p. Taking the product over all p, and using the fact that the sum
of the local degrees at p equals the global degree ŒK W(cid:81)(cid:141), we obtain the desired
globalconclusion. (cid:3)
InordertoderiveacharacterizationofT DQ T forarbitrarynumberfieldsK
K p p
similarto(2),weobservethatwehaveanexactdivisibility`k k#T oftheorder
p
ofT byaprimepower`k ifandonlyifthelocalfieldK atpcontainsaprimitive
p p
`k-throotof unity,but not aprimitive`kC1-throotof unity. Wemayrewordthis
as: TheprimepsplitscompletelyinthecyclotomicextensionK (cid:26)K.(cid:16) /,butnot
`k
inthecyclotomicextensionK (cid:26)K.(cid:16) /. Ifsuchpexistatallfor`k,thenthere
`kC1
areinfinitelymanyofthem,bytheChebotarevdensitytheorem.
Thus, T can be written as a product of groups .(cid:90)=`k(cid:90)/(cid:90) DMap.(cid:90);(cid:90)=`k(cid:90)/
K
that are themselves countable products of cyclic groups of order `k. The prime
powers`k >1thatoccurforK areallbutthoseforwhichwehaveanequality
K.(cid:16) /DK.(cid:16) /:
`k `kC1
ForK D(cid:81)allprimepowers`k occur,butforgeneralK,therearefinitely many
primepowersthatmaydisappear. Thisisduetothefactthattheinfinitecyclotomic
extension(cid:81)(cid:26)(cid:81).(cid:16)`1/withgroup(cid:90)(cid:3)` canpartially“collapse”overK.
26 ATHANASIOSANGELAKISANDPETERSTEVENHAGEN
Todescribetheexceptionalprimepowers`k thatdisappearforK,weconsider,
for`anodd prime,thenumber
w.`/DwK.`/D#(cid:22)`1.K.(cid:16)`//
of`-powerrootsofunityinthefieldK.(cid:16) /. Foralmostall`,thisnumberequals`,
`
andwecall`exceptionalforK ifitisdivisibleby`2. Notethatnooddexceptional
primenumbersexistforimaginaryquadraticfieldsK.
Fortheprime`D2,weconsiderinsteadthenumber
w.2/DwK.2/D#(cid:22)21.K.(cid:16)4//
of 2-power roots in K.(cid:16) /DK.i/. If K contains i D(cid:16) , or if w.2/ is divisible
4 4
by8,wecall2 exceptional forK. NotethatptheonlyimaginaryquadraticfieldsK
forwhich2isexceptionalare(cid:81).i/and(cid:81). (cid:0)2/.
Thenumberw.K/ofexceptionalrootsofunityforK isnowdefinedas
Y
w.K/D w.`/:
`exceptional
Note that w.K/ refers to roots of unity that may or may not be contained in K
itself,andthateveryprime`dividingw.K/occurswithexponentatleast2. The
primepowers`k >1thatdonot occurwhenT iswrittenasadirectproductof
K
groups.(cid:90)=`k(cid:90)/(cid:90) arethestrict divisorsofw.`/atexceptionalprimes`,withthe
exceptionalprime`D2givingrisetoaspecialcase.
Lemma3.2. LetK beanumberfield, andw Dw.K/itsnumberofexceptional
rootsofunity. Thenwehaveanoncanonicalisomorphismofprofinitegroups
Y Y
T D T Š (cid:90)=nw(cid:90);
K p
p n(cid:21)1
exceptwhen2isexceptionalforK andi D(cid:16) isnotcontainedinK. Inthisspecial
4
case,wehave
Y Y
T D T Š .(cid:90)=2(cid:90)(cid:2)(cid:90)=nw(cid:90)/:
K p
p n(cid:21)1
ThegroupTK isisomorphictothegroupT(cid:81) in(2)ifandonlyifwehavewD1.
Proof. If`isodd,thetoweroffieldextensions
K.(cid:16) /(cid:26)K.(cid:16) /(cid:26)(cid:1)(cid:1)(cid:1)(cid:26)K.(cid:16) /(cid:26)K.(cid:16) /(cid:26)(cid:1)(cid:1)(cid:1)
` `2 `k `kC1
isa(cid:90) -extension,andthestepsK.(cid:16) /(cid:26)K.(cid:16) /withk(cid:21)1inthistowerthat
` `k `kC1
areequalitiesareexactlythoseforwhich`kC1 dividesw.`/.
Similarly,thetoweroffieldextensions
K.(cid:16) /(cid:26)K.(cid:16) /(cid:26)(cid:1)(cid:1)(cid:1)(cid:26)K.(cid:16) /(cid:26)K.(cid:16) /(cid:26)(cid:1)(cid:1)(cid:1)
4 8 2k 2kC1
IMAGINARYQUADRATICFIELDSWITHISOMORPHICABELIANGALOISGROUPS 27
isa(cid:90) -extensioninwhichthestepsK.(cid:16) /(cid:26)K.(cid:16) /withk(cid:21)2thatareequal-
2 2k 2kC1
itiesareexactlythoseforwhich2kC1 dividesw.2/. TheextensionK DK.(cid:16) /(cid:26)
2
K.(cid:16) / that we have in the remaining case k D1 is an equality if and only if K
4
containsi D(cid:16) .
4
Thus,aprimepower`k >2thatdoesnotoccurwhenT iswrittenasaproduct
K
ofgroups.(cid:90)=`k(cid:90)/(cid:90) isthesameasastrict divisor`k >2ofw.`/atanexceptional
prime `. The special prime power `k D2 does not occur if and only if i D(cid:16) is
4
inK. Notethatinthiscase,2isbydefinitionexceptionalforK.
ItisclearthatreplacingthegroupQ (cid:90)=n(cid:90)from(2)byQ (cid:90)=nw(cid:90)has
n(cid:21)1 n(cid:21)1
theeffectofremovingcyclicsummandsoforder`k with`kC1jw,andthisshows
thatthegroupsgivenintheLemmaareindeedisomorphictoT . OnlyforwD1
K
weobtainthegroupT(cid:81) inwhichallprimepowers`k arise. (cid:3)
Lemmas3.1and3.2telluswhatOy(cid:3) lookslikeasa(cid:90)y-module. Inparticular,it
showsthatthedependenceonK islimitedtothedegreeŒK W(cid:81)(cid:141),whichisreflected
intherankof thefree(cid:90)y-partofOy(cid:3),andthe natureoftheexceptionalrootsofunity
for K. For the group Oy(cid:3)=(cid:22) , the same is true, but the proof requires an extra
K
argument,andthefollowinglemma.
Lemma3.3. ThereareinfinitelymanyprimespofK forwhichwehave
gcd.#(cid:22) ;#T =#(cid:22) /D1:
K p K
Proof. For every prime power `k > 1 that exactly divides #(cid:22) , the extension
K
K D K.(cid:16) / (cid:26) K.(cid:16) / is a cyclic extension of prime degree `. For different
`k `kC1
primepowers`k k#(cid:22) ,wegetdifferentextensions,soinfinitelymanyprimesp
K
ofK areinertinallofthem. Forsuchp,wehavegcd.#(cid:22) ;#T =#(cid:22) /D1. (cid:3)
K p K
Lemma3.4. WehaveanoncanonicalisomorphismT =(cid:22) ŠT .
K K K
Proof. Pickaprimep ofK thatsatisfiestheconditionsofLemma3.3. Then(cid:22)
0 K
embeds as a direct summand in T , and we can write T Š(cid:22) (cid:2)T =(cid:22) as a
p0 p0 K p0 K
product of two cyclic groups of coprime order. It follows that the natural exact
sequence
Y
1(cid:0)! T (cid:0)!T =(cid:22) (cid:0)!T =(cid:22) (cid:0)!1
p K K p0 K
p¤p0
canbesplitusingthecomposedmapT =(cid:22) !T !T !T =(cid:22) . Thismakes
p0 K p0 K K K
Q
T =(cid:22) isomorphictotheproductof T andacyclicgroupforwhichthe
K K p¤p0 p
orderisaproductofprimepowersthatalready“occur”infinitelyofteninT . Thus
K
T =(cid:22) isisomorphictoaproductofexactlythesamegroups.(cid:90)=`k(cid:90)/(cid:90) thatoccur
K K
inT ,andthereforeisomorphictoT itself. (cid:3)
K K
ForimaginaryquadraticK,whereOy(cid:3)=(cid:22) constitutestheinertialpartU ofA
K K K
from(4),wesummarizetheresultsofthissectioninthefollowingway.
28 ATHANASIOSANGELAKISANDPETERSTEVENHAGEN
Theorem3.5. LetK beanimaginaryquadraticfieldp. ThenthesubgroupTK=(cid:22)K
ofU isadirectsummandofU . ForK ¤(cid:81).i/;(cid:81). (cid:0)2/,wehaveisomorphisms
K K
1
U DOy(cid:3)=(cid:22) Š(cid:90)y2(cid:2).T =(cid:22) /Š(cid:90)y2(cid:2) Y (cid:90)=n(cid:90)
K K K K
nD1
ofprofinitegroups.
p
ForK equalto(cid:81).i/or(cid:81). (cid:0)2/,theprime2isexceptionalforK,andthegroups
T =(cid:22) ŠT aredifferentastheydonothavecyclicsummandsoforder2and4,
K K K
respectively.
4. ExtensionsofGaloisgroups
Intheprevioussection, allresultscouldeasily bestated andprovedforarbitrary
numberfields. Fromnow on,K willdenote animaginary quadraticfield. Inorder
todescribethefullgroupA from(3),weconsidertheexactsequence
K
1(cid:0)!U DOy(cid:3)=(cid:22) (cid:0)!A DKy(cid:3)=K(cid:3)(cid:0) !Cl (cid:0)!1 (6)
K K K K
that describes the class group Cl of K in idelic terms. Here maps the class
K
of the finite idele .xp/p 2 Ky(cid:3) to the class of its associated ideal Qppep, with
e Dord x .
p p p
Thesequence(6)showsthatU isanopensubgroupofA ofindexequaltothe
K K
classnumberh ofK. InviewofTheorem3.5,thisimmediatelyyieldsOnabe’s
K
discoverythatdifferentK canhavethesameabsoluteabelianGaloisgroup.
p
Theorem4.1. AnimaginaryquadraticnumberfieldK ¤(cid:81).i/;(cid:81). (cid:0)2/ofclass
number1hasabsoluteabelianGaloisgroupisomorphicto
G D(cid:90)y2(cid:2) Y (cid:90)=n(cid:90):
n(cid:21)1
InOnabe’s paper[10, §5], thegroup G, whichis notexplicitlygivenbutcharac-
terized by its infinitely many Ulm invariants, is referred to as ‘of type A’. We
will refer to G as the minimal Galois group, as evepry absolute abelian Galois
group of an imaginary quadratic field K ¤ (cid:81).i/;(cid:81). (cid:0)2/ contains a subgroup
isomorphic to G. We will show that there are actuallymany more K having this
absoluteabelianGaloisgroupthanthesevenfieldsK ofclassnumber1towhich
theprecedingtheoremapplies.
Now take for K any imaginary quadratic field of class number h >1. Then
K
Theorem3.5andthesequence(6)showthatA isanabeliangroupextensionof
K
Cl bytheminimalGaloisgroupG fromTheorem4.1. Iftheextension(6)were
K
split, we would find that A is isomorphic to G(cid:2)Cl ŠG; but it turns out that
K K
splittingatthislevelnever occursfornontrivialCl ,inthefollowingstrongsense.
K