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

109 pages
Model Theory of AbsoluteGalois GroupsDissertationzur Erlangung des Doktorgradesder Fakultat˜ fur˜ Mathematik und Physikder Albert{Ludwigs{Universitat˜ Freiburgvorgelegt vonNina FrohnFreiburg, Januar 2011Dekan: Prof. Dr. Kay Konigsmann˜Erster Gutachter: Dr. Jochen KoenigsmannZweiter Gutachter: Prof. Dr. Martin ZieglerDatum der Promotion: 10.12.2010Abteilung fur Mathematische Logik˜Fakultat fur Mathematik und Physik˜ ˜Albert-Ludwigs-Universitat Freiburg˜Eckerstra…e 179104 Freiburg i. Br.IntroductionGaloistheoryisalivelyareaofmathematicalresearchinwhichseveralbranchesof mathematics (as algebra, number theory and geometry) are involved. Thisthesis is concerned with the model theoretic aspects of Galois theory, and inparticular with the model theoretic aspects of the universal object of Galoistheory, the absolute Galois group.Model theory often studies elementary classes of flrst-order structures, or aclassofflrst-orderstructuressharingsomemodel-theoreticproperty.Thecentralquestion in the background of this work is the question whether or not theclass of absolute Galois groups is an elementary class in an appropriate flrst-order language. The answer is likely to be negative: The general group theoreticstructureofabsoluteGaloisgroupsisnotknownverywell,andthereisnogrouptheoretical characterization of those proflnite groups which occur as absoluteGalois groups, not even conjecturally.
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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’ 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. ¥
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