**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Absolute Galois group

Summary

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. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group.
(When K is a perfect field, Ksep is the same as an algebraic closure Kalg of K. This holds e.g. for K of characteristic zero, or K a finite field.)
The absolute Galois group of an algebraically closed field is trivial.
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:R] = 2.
The absolute Galois group of a finite field K is isomorphic to the group
(For the notation, see Inverse limit.)
The Frobenius automorphism Fr is a canonical (topological) generator of GK. (Recall that Fr(x) = xq for all x in Kalg, where q is the number of elements in K.)
The absolute Galois group of the field of rational functions with complex coefficients is free (as a profinite group). This result is due to Adrien Douady and has its origins in Riemann's existence theorem.
More generally, let C be an algebraically closed field and x a variable. Then the absolute Galois group of K = C(x) is free of rank equal to the cardinality of C. This result is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden using algebraic methods.
Let K be a finite extension of the p-adic numbers Qp. For p ≠ 2, its absolute Galois group is generated by [K:Qp] + 3 elements and has an explicit description by generators and relations. This is a result of Uwe Jannsen and Kay Wingberg. Some results are known in the case p = 2, but the structure for Q2 is not known.
Another case in which the absolute Galois group has been determined is for the largest totally real subfield of the field of algebraic numbers.
No direct description is known for the absolute Galois group of the rational numbers.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related people (1)

Related publications (14)

Related concepts (16)

In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group. Every finite extension of a finite field is a cyclic extension.

In algebra, a field k is perfect if any one of the following equivalent conditions holds: Every irreducible polynomial over k has distinct roots. Every irreducible polynomial over k is separable. Every finite extension of k is separable. Every algebraic extension of k is separable. Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power. Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x ↦ x^p is an automorphism of k.

Related courses (7)

Related lectures (21)

MATH-482: Number theory I.a - Algebraic number theory

Algebraic number theory is the study of the properties of solutions of polynomial equations with integral coefficients; Starting with concrete problems, we then introduce more general notions like alg

MATH-317: Algebra V - Galois theory

Galois theory aims at describing the algebraic symmetries of fields. After reviewing the basic material (from the 2nd year course "Ring and Fields") and in particular the Galois correspondence, we wi

MATH-315: Topological groups

We study topological groups. Particular attention is devoted to compact and locally compact groups.

An integer program (IP) is a problem of the form $\min \{f(x) : \, Ax = b, \ l \leq x \leq u, \ x \in \Z^n\}$, where $A \in \Z^{m \times n}$, $b \in \Z^m$, $l,u \in \Z^n$, and $f: \Z^n \rightarrow \Z$ is a separable convex objective function.
The problem o ...

Galois Theory: Dedekind Rings

Explores Galois theory with a focus on Dedekind rings and their unique factorization of fractional ideals.

Galois Theory: Solvability and Radical Extensions

Explores solvability by radicals in Galois theory and the Galois/Abel criterion for solvability.

Dedekind Rings: Theory and Applications

Explores Dedekind rings, integral closure, factorization of ideals, and Gauss' Lemma.

Cohomological Invariants for G-Galois Algebras and Self-Dual Normal Bases. We define degree two cohomological invariants for $G$-Galois algebras over fields of characteristic not 2, and use them to give necessary conditions for the existence of a self--dua ...

2017We use Masser's counting theorem to prove a lower bound for the canonical height in powers of elliptic curves. We also prove the Galois case of the elliptic Lehmer problem, combining Kummer theory and Masser's result with bounds on the rank and torsion of ...