Frobenius element

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic $p$ , an important class which includes finite fields. The endomorphism maps every element to its $p$ -th power. In certain contexts it is an automorphism, but this is not true in general.

Let $R$ be a commutative ring with prime characteristic $p$ (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism F is defined by

for all r in R. It respects the multiplication of R:

and $F (1)$ is clearly 1 also. What is interesting, however, is that it also respects the addition of $R$ . The expression $(r + s) p$ can be expanded using the binomial theorem. Because $p$ is prime, it divides $p!$ but not any $q!$ for $q < p$ ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients

if $1 \leq k \leq p - 1$ . Therefore, the coefficients of all the terms except $r p$ and $s p$ are divisible by $p$ , the characteristic, and hence they vanish. Thus

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