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Concept# Frobenius endomorphism

Summary

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 1 as well. Moreover, 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 r^p and s^p are divisible by p, and hence they vanish. Thus
This shows that F is a ring homomorphism.
If φ : R → S is a homomorphism of rings of characteristic p, then
If FR and FS are the Frobenius endomorphisms of R and S, then this can be rewritten as:
This means that the Frobenius endomorphism is a natural transformation from the identity functor on the category of characteristic p rings to itself.
If the ring R is a ring with no nilpotent elements, then the Frobenius endomorphism is injective: F(r) = 0 means r^p = 0, which by definition means that r is nilpotent of order at most p. In fact, this is necessary and sufficient, because if r is any nilpotent, then one of its powers will be nilpotent of order at most p. In particular, if R is a field then the Frobenius endomorphism is injective.
The Frobenius morphism is not necessarily surjective, even when R is a field. For example, let K = Fp(t) be the finite field of p elements together with a single transcendental element; equivalently, K is the field of rational functions with coefficients in Fp.

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