Frobenius map

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

We write $F:x\mapsto x^{p}$ and note that in characterstic p we have $(x+y)^{p}=x^{p}+y^{p}$ 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 $\mathbf {F} _{p}(X)$, which has as image the proper subfield $\mathbf {F} _{p}(X^{p})$.

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.