In mathematics, a Galois group is a group associated with a certain type of field extension. The study of field extensions (and polynomials which give rise to them) via Galois groups is called Galois theory after Évariste Galois who first invented them.
For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.
Suppose that E is an extension of the field F (written as E/F and read E over F)...
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In mathematics, a Galois group is a group associated with a certain type of field extension. The study of field extensions (and polynomials which give rise to them) via Galois groups is called Galois theory after Évariste Galois who first invented them.
For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.
Suppose that E is an extension of the field F (written as E/F and read E over F). Consider the set of all automorphisms of E/F (that is, isomorphisms α from E to itself such that α(x) = x for every x in F). This set of automorphisms with the operation of function composition forms a group, sometimes denoted by Aut(E/F).
If E/F is a Galois extension, then Aut(E/F) is called the Galois group of (the extension) E over F, and is usually denoted by Gal(E/F).
In the following examples F is a field, and C, R, Q are the fields of complex, real, and rational numbers, respectively. The notation F(a) indicates the field extension...
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