# Frobenius map

From Citizendium

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.