Frobenius map

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In algebra, the Frobenius map is the p-th power map considered as acting on commutative algebras or fields of prime characteristic p.

We write and note that in characterstic p we have so that F is a ring homomorphism. 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 applied to the rational function field , which has as image the proper subfield .

Frobenius automorphism

When F is surjective as well as injective, it is called the Frobenius automorphism. One important instance is when the domain is a finite field.