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Galois Groups

Dissertation

zur Erlangung des Doktorgrades

der Fakultat˜ fur˜ Mathematik und Physik

der Albert{Ludwigs{Universitat˜ Freiburg

vorgelegt von

Nina Frohn

Freiburg, Januar 2011Dekan: Prof. Dr. Kay Konigsmann˜

Erster Gutachter: Dr. Jochen Koenigsmann

Zweiter Gutachter: Prof. Dr. Martin Ziegler

Datum der Promotion: 10.12.2010

Abteilung fur Mathematische Logik˜

Fakultat fur Mathematik und Physik˜ ˜

Albert-Ludwigs-Universitat Freiburg˜

Eckerstra…e 1

79104 Freiburg i. Br.Introduction

Galoistheoryisalivelyareaofmathematicalresearchinwhichseveralbranches

of mathematics (as algebra, number theory and geometry) are involved. This

thesis is concerned with the model theoretic aspects of Galois theory, and in

particular with the model theoretic aspects of the universal object of Galois

theory, the absolute Galois group.

Model theory often studies elementary classes of ﬂrst-order structures, or a

classofﬂrst-orderstructuressharingsomemodel-theoreticproperty.Thecentral

question in the background of this work is the question whether or not the

class of absolute Galois groups is an elementary class in an appropriate ﬂrst-

order language. The answer is likely to be negative: The general group theoretic

structureofabsoluteGaloisgroupsisnotknownverywell,andthereisnogroup

theoretical characterization of those proﬂnite groups which occur as absolute

Galois groups, not even conjecturally. The results in this thesis can hopefully

be used to give a negative answer to this question in the future.

Having this question as a central motivation in mind, this thesis deals with the

following problems:

† Which ﬂrst order language should be chosen if one wants to set up a model

theory of absolute Galois groups, or more general of proﬂnite groups? The

answer to this question is not as obvious as one might think: The usual group

language may come to one’s mind ﬂrst, but it has the disadvantage of not

capturing the topological structure of the group. An approach that is bet-

ter adapted to proﬂnite groups is describing the inverse system of the group

instead of the group itself.

† Whichsubclassesoftheclassofallproﬂnitegroupsareelementary?Forexam-

ple, in Chapter 5 we give a complete classiﬂcation of the elementary theories

of abelian proﬂnite groups.

† Which properties shared by all absolute Galois groups are axiomatizable, and

what is the elementary theory of the class of absolute Galois groups? We will

investigate various properties of absolute Galois groups and check them for

axiomatizability.

† Are there natural subclasses of the class of all absolute Galois groups from

which we can prove or disprove that they are elementary? For example we

will show that for every ﬂxed natural number n, the class of maximal pro-p

quotients of absolute Galois groups of rank n is elementary, see Corollary 3.7.

Intheeighties,G.Cherlin,L.vandenDriesandA.Macintyreintroducedaﬂrst-

order language in which they considered proﬂnite groups as model theoretic

istructures, see [CDM]. Their attempt was based on substituting the original

proﬂnite group by its associated inverse system, and describing this in an ap-

propriateﬂrst-orderlanguage.Thepreprint[CDM]containsalreadyﬂrstresults

about absolute Galois groups, for example that the absolute Galois group of K

isinterpretableinK,andthattheclassofabsoluteGaloisgroupsisclosedunder

ultraproducts. There is later and more extensive work on the model theory of

proﬂnite groups by Z. Chatzidakis, see for example [C1] and [C2].

Buttothisday,themodeltheoryofproﬂnitegroupsisnotveryelaborated,and

so big parts of this thesis are concerned with the development of the model the-

ory of arbitrary proﬂnite groups. An example: To work towards a proof for the

conjecture that the class of absolute Galois groups is non-elementary, we need

to understand the conditions under which absolute Galois groups are elemen-

tary substructures of each other. For this purpose, we ﬂrst need to understand

elementary equivalence in arbitrary proﬂnite groups.

Other parts of this thesis are concerned with the algebraic structure of abso-

lute Galois groups not including any model theoretic content. The reason is

the following: Studying the model theory of absolute Galois groups often calls

one’s attention to questions concerning the algebraic analysis of absolute Ga-

lois groups. Sometimes solving this algebraic question is just a necessary duty

before proceeding to the model theoretic problem behind it, but sometimes the

algebraic question is worth a discussion on its own sake. An example is Chap-

ter 4 about Demushkin groups: The study of these groups in connection with

the questions about axiomatizability in Section 3.4 eventually led to the purely

algebraic results stated in Chapter 4.

Finally,thetwoaspects\necessarypreliminarywork"and\interestingsideprod-

uct"canmelttogetherasinChapter 5:Thecharacterizationofabelianproﬂnite

groupsgiventhereisintendedtobeusedtoconstructapairG `G ofproﬂnite1 2

groups, such that G is an absolute Galois group, and such that the maximal2

abelian quotient of G guarantees that G cannot be an absolute Galois group,1 1

compare the considerations on page 62. But the classiﬂcation is very interesting

in itself and leads to a satisfying result concerning the model theory of abelian

proﬂnite groups.

This thesis is organized as follows:

In Chapter 1, the necessary foundations for the thesis are provided: Basics

about proﬂnite groups, with a special emphasis on the cohomology of proﬂnite

groups and on Brauer groups; some fundamental facts about (abstract) abelian

groups; and ﬂnally some standard valuation theory.

Chapter 2 is concerned with the model theory of arbitrary proﬂnite groups.

We will mostly work in the language L introduced in [CDM] describing theIS

inverse system of the proﬂnite group G, but we will also compare the expressive

power of L with the oneof the usualgrouplanguage L . MoreoverwedevelopIS G

a characterization of L -elementary equivalence of proﬂnite groups by meansIS

of a modiﬂed Ehrenfeucht-Fraiss¶e game which will be used in Chapter 5.

InChapter 3, we will discuss the axiomatizability of some properties P shared

byallabsoluteGaloisgroups.Thedi–cultyofthisprocedureisthattheproperty

P might not be deﬂned by group theoretical means, but using ﬂeld theoretical

concepts. Thus before trying to axiomatize P, we have to exchange it with a

0propertyP thatisascloseaspossibletoP,butwhichisdeﬂnedbygrouptheo-

iireticalmeans.WediscussindetailthepropertyofabsoluteGaloisgroupsstated

in the Artin-Schreier Theorem (see Section 3.10), and the property formulated

in the so-called\Elementary Type Conjecture"(see Section 3.4). Moreover we

discuss whether or not the cyclotomic quotient of an absolute Galois group is

ﬂrst-order deﬂnable, see Section 3.3.

Chapter4isofpurelyalgebraicnature.Wediscussso-calledDemushkingroups,

examples of which are the maximal pro-p-quotients of absolute Galois groups

of p-adic ﬂelds. First we develop the necessary tools needed in Section 3.4,

where Demushkin groups play a major role. Further we analyze the structure

of Demushkin groups which are realized as maximal pro-p-quotients of absolute

Galois groups.

Finally, in Chapter 5, we give a classiﬂcation of abelian proﬂnite groups up

to elementary equivalence, and prove some model theoretic results about cate-

goricity, stability and the structure of saturated models.

Acknowledgements:

In the ﬂrst place, I would like to thank my supervisor Dr. Jochen Koenigsmann

for suggesting me this topic and for many and considerable discussions which

he managed to arrange even under unfavorable geographic conditions. Without

his advice, this dissertation would not have been possible.

Secondly, I would like to thank Prof. Martin Ziegler: Without the discussions

with him, the last chapter of this thesis would not exist. Moreover, he often

oﬁered mathematical ﬂrst-help when I was hopelessly stuck in the twists and

turns of my confused brain.

Many others made valuable contributions. In particular, I am grateful to Dr.

MarkusJunkerforhismathematical,butalsoemotionalsupport.Finally,Iwant

toapologizetoallthosewhosuﬁeredfrommymoaningandmychangingmoods

in the past few years.

iiiContents

Introduction i

1 Preliminaries 1

1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Proﬂnite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Cohomology of proﬂnite groups and Brauer groups . . . . . . . . 7

1.4 Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Valuation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Model theory of proﬂnite groups 21

2.1 Many-sorted logics . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 The complete inverse system of a proﬂnite group . . . . . . . . . 23

2.3 Elementary equivalence of proﬂnite

groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 The expressive power of L . . . . . . . . . . . . . . . . . . . . . 31IS

3 Model theory of absolute Galois groups 37

3.1 Model theory of absolute Galois groups . . . . . . . . . . . . . . 37

3.2 Formalizations of the Artin-Schreier theorem . . . . . . . . . . . 39

3.3 Searching for the cyclotomic quotient or\dead ends’ death" . . . 46

3.4 The Elementary Type Conjecture . . . . . . . . . . . . . . . . . . 52

3.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Demushkin groups 63

4.1 Basics about Demushkin groups . . . . . . . . . . . . . . . . . . . 63

4.2 Demushkin groups as Galois groups. . . . . . . . . . . . . . . . . 65

4.3 Demushkin groups as semidirect products . . . . . . . . . . . . . 73

4.4 An alternative deﬂnition of Demushkin Galois groups. . . . . . . 74

5 A classiﬂcation of proabelian groups 775.1 A ﬂrst classiﬂcation . . . . . . . . . . . . . . . . . . . . . . . . . 78

⁄5.2 Connecting A, A and S(A) . . . . . . . . . . . . . . . . . . . . . 87

5.3 Some model theory of proabelian groups . . . . . . . . . . . . . . 91Chapter 1

Preliminaries

1.1 Notations

For simplicity, we will use the symbol A both for the domain of a ﬂrst order

structure (A;:::) and for the structure A itself. Moreover, we will not distin-

guishnotationallybetweenthesymbolsofalanguageLandtheirinterpretation

inagivenstructure.If AisanL-structureforsomeﬂrst-orderlanguageLandif

C?A is a parameter set, we denote by A the expansion of A to the languageC

L which arises from L by adding new constant symbols for the elements of C.C

Analogously we deﬂne A for a tuple c„of parameters. If A;B are L-structuresc„

and if C is a common subset of A and B, we write A · B if the expandedC C

structures are elementarily equivalent. We write Th(A), or Th (A) if the lan-L

guage L is not clear from the context, for the ﬂrst-order theory of a structure

A. For a formula ’, the notation ’(x ;:::;x ) indicates that the free variables1 n

of ’ are among x ;:::;x ; similarly for a type p(x ;:::;x ). If the length of1 n 1 n

the tuple is not of relevance, we sometimes write x„ for a tuple (x ;:::;x ).1 n

Sometimeswedenotewith x„ alsotheresidueclassofanelement xmodulosome

equivalence relation; it will be clear from the context which is the at a time

intended meaning. Some notations from the algebraic context, which we will

use in the sequel without further explanation, are:

P the prime numbers

C the complex numbers

F the ﬂeld with q elementsq

Z the (group or ring) of p-adic integersp

id the identity map on a set AA

f the restriction of the map f to the set AjA

ker(’), im(’) kernel and image of the map ’

Z=n the cyclic group with n elements

ord (g) or ord(g) the order of the element g in the group GG

[G:H] the index of the subgroup H in the group G

¡1 ¡1[x;y] the commutator x y xy of x;y

char(K) the characteristic of the ﬂeld K

1£K the multiplicative group of the ﬂeld K

n‡ n a primitive p -th root of unity (in the current ﬂeld)p

„ (K) the set of all n-th roots of unity contained in K.n

i„ 1(K) the set of all p -th roots of unity contained in K, i‚1.p

„ 1 =„ 1(C) the Prufer group (belonging to the prime p).˜p p

Gal(L=K) the Galois of a Galois ﬂeld extension L=K

G the absolute Galois group of the ﬂeld KK

aK an algebraic closure of the ﬂeld K

sK a separabel algebraic closure of the ﬂeld K

N the norm map of an algebraic ﬂeld extension L=KL=K

We will denote both the trivial group and the neutral element of a group with

1, or with 0 in the abelian case.

1.2 Proﬂnite groups

In this section, we brie y recall the facts about proﬂnite groups that we will

need. The presentation follows [RZ] and [Wi].

1.2.1 Fundamentals and Conventions

Recall that a proﬂnite group is an inverse limit of an inverse system of ﬂnite

groups;equivalently,itisacompact,Hausdorﬁ,totallydisconnectedtopological

group. A pro-C-group for a suitable classC of ﬂnite groups is an inverse limit of

groups inC; we are mainly interested in the case thatC is the class of all ﬂnite

groups (leading to arbitrary proﬂnite groups), or of all ﬂnite p-groups (leading

to pro-p-groups).

Ifwearedealingwithproﬂnitegroupsandifnototherwisestated,

we do always assume a topological background.

This means: Subgroups are assumed to be closed, group homomorphisms to be

continuous, isomorphisms to be homeomorphisms etc. Consequently we write

U • G for a closed subgroup U of G, and N C G if N is a closed normal

subgroup. The symbols • ;C indicate that we are dealing with open (respec-o o

tively open normal) subgroups; recall that open subgroups of proﬂnite groups

are in particular closed. If H is a closed subgroup of G, then H with the sub-

group topology inherited from G is proﬂnite; in particular, if (N j N 2 I)

»is a family of open normal subgroups of G such that G lim G=N, then= ˆ¡N2I

»H = lim H=(H \N). If G ‡ H is a continuous epimorphism of proﬂnite

N2Iˆ¡

groups, then H is equipped with the quotient topology. A quotient of G modulo

aclosednormalsubgroupiscalleda continuous quotient.ThecartesianproductQ

G ofproﬂnitegroupsG isaproﬂnitegroupwhenequippedwiththeproducti i

topology.

IfG is proﬂnite and X is a subset of G, we denote by X its closure in G, and by

hXitheclosedsubgroupgeneratedbyX;furtherhjXjidenotesthesmallestclosed

normal subgroup of G containing X. We say that X generates G if hXi = G;

2further a set X converges to 1 if every open subgroup of G contains all but a

ﬂnitenumberoftheelementsinX.Therank ofG,whichwedenotebyrank(G),

is the smallest cardinality of a set of topological generators of G converging to

1. A proﬂnite group G is generated by a countable 1-convergent set if and only

if there is a chain G = N ‚ N ‚ N ‚ ::: of open normal subgroups of G0 1 2

comprising a base of open neighborhoods of 1.

Aﬂnitelygeneratedproﬂnitegroupisdeterminedbyitsﬂnitecontinuousimages:

Iftwoﬂnitelygeneratedproﬂnitegroupshavethesameﬂnitecontinuousimages,

thentheyareisomorphic.Anopensubgroupofaproﬂnitegrouphasﬂniteindex;

for ﬂnitely generated proﬂnite groups, the converse holds as well. This is not

hard to show for pro-p-groups; the general case is a recent and sophisticated

result by N. Nikolov and D. Segal, see [NS]. In arbitrary proﬂnite groups, one

only knows that open subgroups are closed, and that closed subgroups are open

if and only if they have ﬂnite index.

A proﬂnite group of rank • has a (free) presentation G = F=N, where F is a

free proﬂnite group of rank • and N is a closed normal subgroup. If G is pro-p,

there is also such a presentation where F is free pro-p of rank •.

1.2.2 The Frattini subgroup

Deﬂnition 1.1. For a proﬂnite group G, deﬂne the Frattini subgroup '(G) to

be the intersection of all proper maximal open subgroups of G. For i‚0, deﬂne

0 i+1 i 2inductively ' (G) = G and ' (G) = '(' (G)); the series '(G)‚ ' (G)‚

i i::: is called the Frattini series of G. Further for i‚1 letF (G):=G=' (G) be

the i-th Frattini quotient of G.

Inthesequel,wedenoteby [G;G]thederivedsubgroupofG;thisistheabstract

subgroup of G generated by the commutators. If G is ﬂnitely generated, then

p[G;G] is a closed subgroup. Further G is the abstract subgroup of G generated

by the p-th powers of G.

Lemma 1.2. Let G;H be pro-p-groups.

1) '(G) is open if and only if G is ﬂnitely generated.

p2) '(G)=G [G;G].

3) Every continuous epimorphism ’:G!H restricts to an epimorphism from

i i i i' (G) to ' (H) and induces a continuous epimorphism ’:F (G)!F (H).

If G and H are moreover ﬂnitely generated, we further have

i4) All ' (G) are open normal subgroups of G.

p5) '(G)=G [G;G].

6) The Frattini series constitutes a base of open neighborhoods of 1. Thus if G

has the same Frattini quotients as H, then G and H are isomorphic.

rank(G) rank(G)»7) G='(G)=(Z=p) ; in particular, [G:'(G)]=p .

Corollary 1.3. If G is ﬂnitely generated and H •G is a closed subgroup, then

H has countable rank.

Proof: The intersections H \ ' (G) form a countable base of open neigh-i

borhoods of 1 in H; thus H is generated by a countable 1-convergent set, cf.

Subsection 1.2.1. ¥

3