Definable Henselian Valuations and Absolute Galois Groups

105 Pages·2014·0.65 MB·English

University of Oxford
Mathematical Institute
Lady Margaret Hall
Definable Henselian Valuations
and
Absolute Galois Groups
A thesis submitted for the degree of
Doctor of Philosophy
Author: Supervisor:
Franziska Jahnke Dr Jochen Koenigsmann
Hilary 2014

Abstract
This thesis investigates the connections between henselian valuations and absolute Galois
groups. There are fundamental links between these: On one hand, the absolute Galois group
of a field often encodes information about (henselian) valuations on that field. On the other,
in many cases a henselian valuation imposes a certain structure on an absolute Galois group
which makes it easier to study.
We are particularly interested in the question of when a field admits a non-trivial parameter-
free definable henselian valuation. By a result of Prestel and Ziegler, this does not hold for
every henselian valued field. However, improving a result by Koenigsmann, we show that
there is a non-trivial parameter-free definable valuation on every henselian valued field. This
allows us to give a range of conditions under which a henselian field does indeed admit a non-
trivial parameter-free definable henselian valuation. Most of these conditions are in fact of a
Galois-theoretic nature. Since the existence of a parameter-free definable henselian valuation
on a field ensures that henselianity is elementary in L , we also study henselianity as an
ring
L -property.
ring
Throughout the thesis, we discuss a number of applications of our results. These include
fields elementarily characterized by their absolute Galois group, model complete henselian
fields and henselian NIP fields of positive characteristic, as well as PAC and hilbertian fields.
Acknowledgements
Iwouldliketothankmysupervisor, JochenKoenigsmann, forhispatience, helpandsupport.
There is no doubt that this thesis would not have been possible without him. His constant
enthusiasm and our regular meetings kept me motivated over the last three years. I couldn’t
have had a better supervisor.
I would further like to thank MALOA, that is the organizers and institutions behind it, for
both the generous funding and the endless opportunities which being a MALOA student
entailed.
My thanks also goes to the Logic group in Oxford. In particular, I would like to thank Boris
Zilber and Jonathan Pila, for examining and commenting on earlier versions of this thesis.
Special thanks goes to Will, Bernhard, Lotte, Ugur and Austin, for their excellent company
and willingness to help and proof read. Thanks to the Logic group, I had a great time, many
good dinners and the chance to learn much more Model Theory.
I would like to thank Tom Scanlon as well as Katrin Tent for the productive visits to their
respective research groups. My thanks also goes to Arno Fehm, Immanuel Halupzok and Itay
Kaplan for their interest, help and comments.
Last but not least, I’d like to thank Tobi, my family and Sigrid for their love, support and
company.

Contents
Introduction v
The Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Galois Theory and Valued Fields 1
1.1 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The Absolute Galois Group . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Sylow Subgroups of Profinite Groups . . . . . . . . . . . . . . . . . . . 3
1.1.3 Profinite Groups as Elementary Structures . . . . . . . . . . . . . . . 4
1.1.4 Projective Profinite Groups . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Hilbert Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Henselian Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Henselian Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 The Canonical Henselian Valuation . . . . . . . . . . . . . . . . . . . . 13
1.3.3 Finding Henselian Valuations via Galois Groups . . . . . . . . . . . . 15
2 Notions of Henselianity 17
2.1 p-Henselianity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 The Canonical p-Henselian Valuation. . . . . . . . . . . . . . . . . . . 17
2.1.2 Definitions of p-Henselian Valuations . . . . . . . . . . . . . . . . . . . 19
2.1.3 A Uniform Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
i
2.1.4 p-Henselianity and Subfields . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 t-Henselianity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 t-Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Finding a Parameter-Free Definable Valuation . . . . . . . . . . . . . 25
2.2.3 Henselianity as an Elementary Property . . . . . . . . . . . . . . . . . 27
2.3 An Example: PAC and Hilbertian Fields . . . . . . . . . . . . . . . . . . . . . 32
2.3.1 Hilbertian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.2 PAC Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.3 A Separating Sentence . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Definable Henselian Valuations 39
3.1 Conditions on the Residue Field . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 p-Henselian Valuations as Henselian Valuations . . . . . . . . . . . . . 39
3.1.2 Separably Closed, Real Closed, or p-Closed Residue Fields . . . . . . . 40
3.1.3 Not p-Henselian Residue Fields . . . . . . . . . . . . . . . . . . . . . . 42
3.1.4 Not Virtually p-Henselian Residue Fields . . . . . . . . . . . . . . . . 43
3.2 Galois-Theoretic Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Pro-Nilpotent Absolute Galois Groups . . . . . . . . . . . . . . . . . . 45
3.2.2 Non-Universal Absolute Galois Groups . . . . . . . . . . . . . . . . . . 45
3.2.3 Galois-Theoretic Conditions on the Residue Field . . . . . . . . . . . . 48
3.2.4 NIP Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Elimination of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 An Application: Model Complete Fields . . . . . . . . . . . . . . . . . . . . . 53
4 Elementary Characterization 57
4.1 Fields Elementarily Characterized by their Absolute Galois Group . . . . . . 57
4.1.1 Notions of Determination . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.2 The Classification Theorem . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Redefining Classes (D) and (E) . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Defining Henselian Valuations . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.2 Defining a New Class (D*) . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.3 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 An Excursion: Abstract Elementary Classes . . . . . . . . . . . . . . . . . . . 69
4.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.2 The Class K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
G
4.3.3 The Class Kelem and another Notion of Embedding . . . . . . . . . . . 73
G
4.3.4 Amalgamation for (K ,≺ ) . . . . . . . . . . . . . . . . . . . . . . . . 75
G G
Outlook 76
Definable Henselian Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Bibliography 80
Index 85

Introduction
Valued fields play a fundamental role in several different mathematical disciplines, most no-
tably in number theory and algebraic geometry. This has led to considerable model-theoretic
interest in valuations. A valuation assigns a certain size or multiplicity to elements of a field,
and is a generalization of the notion of an absolute value.
At the end of the 19th century, Kurt Hensel was the first to introduce and study the p-adic
numbers, and hence implicitly formed the idea of a valuation ([Hen97]). The first formal
definition of a valuation (with values in the additive group of the real numbers) came from
J´oszef Ku¨rsch´ak ([Ku¨r13]). Wolfgang Krull later developed the notion of a valuation as we
know it today ([Kru30]). From the model-theoretic point of view, allowing arbitrary ordered
abelian value groups is vital – clearly, being a subgroup of the additive group of the real
numbers is not preserved under elementary equivalence in the language of ordered groups.
On the model-theoretic side, valued fields were already studied by Abraham Robinson. He
proved that the theory of algebraically closed valued fields (ACVF) is model complete in the
language of valued fields, i.e. the language of fields together with a symbol for the valuation
ring ([Rob56]). In fact, it follows from his proof that ACVF also eliminates quantifiers in
this language. Since then, a lot of work has been done on the model theory of ACVF; in
particular, using the advanced model-theoretic technique of stable domination ([HHM08]).
There is plenty of ongoing research in this area, not least because model-theoretic results on
ACVF give information about quantifier-free definable sets in any valued field.
Amajorbreakthroughinadifferentareaofthemodeltheoryofvaluedfieldscameinthe1960’s
v