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.