Frobenius endomorphism

Definition

Let $p$ be a prime number and let $R$ be a commutative ring of characteristic $p$. The Frobenius endomorphism of $R$ is the map $F \colon R \to R$, $F(x) = x^p$. It is a ring homomorphism from $R$ to $R$.

Remarks

The map $F$ preserves products because $R$ is commutative. It preserves sums by the freshman's dream. If $R$ is an integral domain, then $F$ is injective. If $R$ is a finite field, then $F$ is an automorphism of $R$. On $\mathbb{F}_p$, the map $F$ is the identity, by Fermat's little theorem.

Explanations

L1 human

In characteristic $p$, raising to the $p$-th power respects addition and multiplication. So it is a homomorphism from the ring to itself. This simple map is a basic tool in the study of finite fields.