Field automorphism

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In field theory, a field automorphism is an automorphism of the algebraic structure of a field, that is, a bijective function from the field onto itself which respects the fields operations of addition and multiplication.

The automorphisms of a given field K form a group, the automorphism group Aut(K).

If L is a subfield of K, an automorphism of K which fixes every element of L is termed an L-automorphism. The L-automorphisms of K form a subgroup AutL(K) of the full automorphism group of K. A field extension K / L of finite index d is normal if the automorphism group is of order equal to d.

[edit] Examples

A homomorphism of fields is necessarily injective, since it is a ring homomorphism with trivial kernel, and a field, viewed as a ring, has no non-trivial ideals. An endomorphism of a field need not be surjective, however. An example is the Frobenius map \Phi: x \mapsto x^p applied to the rational function field \mathbf{F}_p(X), which has as image the proper subfield \mathbf{F}_p(X^p).

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