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Frobenius morphism


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 ≤ k ≤ p − 1. Therefore, the coefficients of all the terms except rp and sp are divisible by p, the characteristic, and hence they vanish. Thus


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