A field automorphism of a field F is a bijective map σ:F->F that preserves all of F's algebraic properties, more precisely, it is an isomorphism. For example, complex conjugation is a field automorphism of C, the complex numbers, because 0^_ | = | 0 1^_ | = | 1 (a + b)^_ | = | a^_ + b^_ (a b)^_ | = | a^_ b^_. A field automorphism fixes the smallest field containing 1, which is Q, the rational numbers, in the case of field characteristic zero.

automorphism | extension field | field | Galois group