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The absolute Galois group of the real numbers is a cyclic group of order 2 generated by complex conjugation, since C is the separable closure of R and [C:R] = 2.. In mathematics, the absolute Galois group G K of a field K is the Galois group of K sep over K, where K sep is a separable closure of K.
One of the most studied classes of infinite Galois group is the absolute Galois group, which is an infinite, profinite group defined as the inverse limit of all finite Galois extensions / for a fixed field. The inverse limit is denoted
Equivalence classes of irreducible ℓ-adic representations σ(π) of dimension n of the absolute Galois group of F; that preserves the L-function at every place of F. The proof of Lafforgue's theorem involves constructing a representation σ(π) of the absolute Galois group for each cuspidal representation π.
The absolute Galois group Gal(Q) (where Q are the rational numbers) is compact, and hence equipped with a normalized Haar measure. For a Galois automorphism s (that is an element in Gal(Q)) let N s be the maximal Galois extension of Q that s fixes. Then with probability 1 the absolute Galois group Gal(N s) is free of countable rank
For p-adic fields the Weil group is a dense subgroup of the absolute Galois group, and consists of all elements whose image in the Galois group of the residue field is an integral power of the Frobenius automorphism. More specifically, in these cases, the Weil group does not have the subspace topology, but rather a finer topology.
The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida (Neukirch–Uchida theorem, 1969), prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck.
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Galois theory implies that, since the polynomial is irreducible, the Galois group has at least four elements. For proving that the Galois group consists of these four permutations, it suffices thus to show that every element of the Galois group is determined by the image of A, which can be shown as follows.