Frobenius homomorphism

Let $F$ be a field of characteristic $p>0$ . Then for any $a, b \in F$ , \begin{eqnarray*} (a + b)^p &=& a^p + b^p, \\ (ab)^p &=& a^p b^p. \end{eqnarray*} Thus the map $$ \begin{matrix}\phi: F &\to& F \\ a &\mapsto& a^p \end{matrix} $$ is a field homomorphism, called the Frobenius homomorphism, or simply the Frobenius map on $F$ . If it is surjective then it is an automorphism, and is called the Frobenius automorphism.

Note: This morphism is sometimes also called the ``small Frobenius'' to distinguish it from the map $a \mapsto a^q$ , with $q=p^n$ . This map is then also referred to as the ``big Frobenius'' or the ``power Frobenius map''.