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Weil–Deligne Group
In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local field, local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite level" modifications of the Galois groups: if ''E''/''F'' is a finite extension, then the relative Weil group of ''E''/''F'' is ''W''''E''/''F'' = ''WF''/ (where the superscript ''c'' denotes the commutator subgroup). For more details about Weil groups see or or . Weil group of a class formation The Weil group of a class formation with fundamental classes ''u''''E''/''F'' ∈ ''H''2(''E''/''F'', ''A''''F'') is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program. If ''E''/''F'' is a normal layer, then the (relative) Weil group ''W''''E''/''F'' of ''E''/''F'' is the extension :1 → ''A''''F'' → ''W''''E''/''F'' → Gal(''E''/''F'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Galois Group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep 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, ''K''sep is the same as an algebraic closure ''K''alg of ''K''. This holds e.g. for ''K'' of characteristic zero, or ''K'' a finite field.) Examples * 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:Rnbsp;= 2. * The absolute Galois group of a finite field ''K'' is isomorphic to the group :: \hat = \varprojlim \mathbf/n\mathbf. (For the notation, see Inverse limit.) :The Frobenius automorphism Fr is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
