Model theory of absolute Galois groups [Elektronische Ressource] / vorgelegt von Nina Frohn

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• Mathematics

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• Publié par: albert-ludwigs-universitat_freiburg
• Publié le: 01 janvier 2011
• Nombre de lectures: 18
• Langue: English

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Model Theory of Absolute
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’