Frobenius homomorphism

Let $F$ be a field of characteristic $p>0$. Then for any $a,b\in F$,

${(a+b)}^p$$=$$a^p+b^p,$

${(ab)}^p$$=$$a^p{b}^p.$

Thus the map

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”.