Pdf the absolute galois group of a hilbertian prc field. Actions of the absolute galois group hal archive ouverte. Map from absolute galois group of a global field to that. This allows us to perform computations in the galois group more simply. And for this question, as far as i know, no book would really meet your ideal, since as you can see, there are too many things about the absolute galois group, everything about number theory is about that group. Alternatively it is the group of all automorphisms of the algebraic closure of k that fix k. Topological realisations of absolute galois groups 5 exponentialsequence 0.
The absolute galois group of an algebraically closed field is trivial. That being said, the structure of the absolute galois group over the rationals which you can think of as a massive group comprised out of all galois groups of finite extensions of rationals, i. Galois cohomology seminar week 6 galois cohomology. F suppressing the choice of base point in the introduction. So it was natural to turn to proversions of the hyperbolic anabelian conjectures, replacing the fundamental groups by their maximal proquotients and the absolute galois group g k by g k. Gabino gonzalezdiez joint work with andrei jaikinzapirain on the action of the absolute galois group on triangle curves. The case k q, or more generally when k is a number field, is the most interesting case in number theory, but is also the most difficult. It is a fundamental problem to classify the absolute galois groups g f of.
Teichmuller group gtd and its action on pro nite braids. Element in the absolute galois group of the rationals. The absolute galois group of a hilbertian prc field article pdf available in israel journal of mathematics 851. The sylow subgroups of the absolute galois group galq. Examples of galois groups and galois correspondences. We present possible extensions of the quantum statistical mechanical formulation of class field theory to the nonabelian case, based on the action of the absolute galois group on grothendiecks dessins denfant, the embedding in the grothendieckteichmuller group, and the drinfeldihara involution.
The dream was formulated after a major result of kroneckerw eber on. In its original form, that conjecture states that the absolute galois group of qab is free, where qab is the maximal abelian extension of q. Detecting propgroups that are not absolute galois groups by dave benson at aberdeen, nicole lemire and ja. Indeed, the absolute galois group gk of every pac field k is projective. But i think you probably want to learn more about galois representation. However, we do know the structure of the absolute galois group of certain distinguished algebraic extensions of q, or, more generally, of a countable hilbertian. Our search for reciprocity laws can be rephrased as the study of a single group, the absolute galois group of the field of rational numbers, written galqq. This group turns out to be realfree, determined up to isomorphism by the.
Braids, galois groups, and some arithmetic functions. We treat g k as a fundamental object of study because it allows us to control all separable. Analogously, given a topological space x, we get a group, its. These notes give a concise exposition of the theory of fields, including the galois theory of finite and infinite extensions and the theory of transcendental. Detecting propgroups that are not absolute galois groups. In mathematics, more specifically in the area of abstract algebra known as galois theory, the galois group of a certain type of field extension is a specific group associated with the field extension.
The absolute galois group of the real numbers is a cyclic group of two elements complex conjugation and the identity map, since c is the separable closure of r and c. Galois cohomology seminar week 6 galois cohomology, absolute galois group, hilbert 90 joshua ruiter february 17, 2019 note on sources the main sources for these notes are sections 2. The galois group gq p acts on cpand is in fact the group of continuous automorphisms of cp. In the last chapter i show how this relates to the absolute galois group by giving an explicit description of the action of the absolute galois group on the fundamental group of a particularly simple moduli space. The absolute galois group of a field k is that of the field extension k. Exploring the galois group of the rational numbers. Cp with absolute value less than or equal to 1 form a closed subring zp resp. The goal of classical langlands program is to understand the galois group of algebraic numbers as algebraic extension of rationals absolute galois group agg through its representations. On function fields with free absolute galois groups by. Let fbe a nite set of classical local elds of characteristic 0 which is closed under galois isomorphism and let gbe a pro nite group. Number theory textbook based on the absolute galois group. Although it is not the simplest it is arguably the most natural galois group to study.
The challenge of all challenges for a number theorist is to understand the galois group of algebraic numbers regarded as extension of rationals by its fundamental importance this group deserves to be called absolute galois group agg, 1. Given a eld k, we get a group, its absolute galois group, whose subgroups we can use to understand extensions of k. In other words, we want to study the structure of the absolute galois group g k of k as a profinite group. Map from absolute galois group of a global field to that of. Since each automorphism in the galois group permutes the roots of 4. The representation theory of galqq has been particularly fruitful in answering instances of question a.
Galois theory for schemes of websites universiteit leiden. The absolute galois group over the rational number field q, denoted by gq galqq, is one of the classical mathematical objects which we really need to understand better. F also has a classical fundamental group, given by homotopy classes of loops. The absolute galois group of a finite field k is isomorphic to the group. We determine the absolute galois group of a countable hilbertian p seudo real closed fieldp of characteristic 0. In abstract algebra, we considered nite galois extensions of elds with their galois groups. The absolute galois group is welldefined up to inner automorphism. Pdf actions of the absolute galois group semantic scholar. The absolute galois group of a pac field is projective ax and haran frj, theorem 10. The absolute galois group of cx school of mathematical. The main aim of this paper is to study the absolute galois group g of. Then the absolute galois group of kis isomorphic to the absolute galois group of fpt. We will also let gq denote the group of automorphisms of q, that is galqq, the absolute galois group of q. The action of dgt on bbn is explained in theorem 1.
The global langlands conjecture sets up a bijection between isomorphism classes of certain representations of gl na and isomorphism classes of certain q valued representations of g q of dimension n. Pro nite groups, pro nite completions and the absolute galois group 6 3. Particularly an action of the absolute galois group of the rational number eld on pro nite knots is rigorously established, which is attained by our extending the action of drinfelds grothendieckteichmuller group on pro nite braid groups into on pro nite knots. In mathematics, the absolute galois group gk of a field k is the galois group of ksep over k, where ksep is a separable closure of k. The action of the absolute galois group on dessins denfants 21 7. Conversely, if g is a projective group and k is a field, then there exists an extension f of k which is pac such that gf. Quantum statistical mechanics of the absolute galois group. Invertible adeles ideles define gl1 which can be shown to be isomorphic with the galois group of maximal abelian extension of rationals. When k is a perfect field this is equivalently the galois group of the algebraic closure k. Theorem drinfeld, ihara, deligne there is an inclusion of the absolute galois group of the rational numbers into the grothendieckteichmuller group recalled e. Faithful actions of the absolute galois group on connected components of moduli spaces. This group is monstrously big since it is in some sense union of all nited galois groups.
Q the absolute galois group of q, a the qad eles and na positive integer. Also, a profinite group is projective if and only if it is isomorphic to a closed subgroup of a free profinite group, and gct is free. I end by showing how this description was used by y. Galois groups and fundamental groups berkeley math. Galois groups and fundamental groups abstract galois theory describes the rich connection between eld theory and group theory. Then gis isomorphic to the absolute galois group of a pfc eld k if and only if gis fprojective and. Here, we noticed a correspondence between the intermediate elds and the subgroups of the galois group.
On the action of the absolute galois group on triangle curves. Theorem c let c be a class of finite groups closed under forming subgroups, quotients and extensions, let gc be the class of absolute galois groups which. Let c be an algebraically closed field of cardinality m. Usually when people talk on the absolute galois group g. The teichmuller tower and the grothendieckteichmuller group 19 6. Localglobal galois theory of arithmetic function fields. Galois groups and fundamental groups by tamas szamuely. Several results in this direction were obtained in. Ingeneral,thesingularcohomologygroupsofx f donotagreewith. Some nonlinear actions of the absolute galois groups 9 4. The absolute galois group of a global field springerlink. The study of field extensions and their relationship to the polynomials that give rise to them via galois groups is called galois theory, so.
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