Frobenius endomorphism

In ordinary arithmetic, (a + b) raised to a power is a messy expansion. But in a world of prime characteristic p, almost all the cross-terms in (a + b)^p are divisible by p and therefore vanish. The result is the 'freshman's dream' come true: (a + b)^p = a^p + b^p. Raising to the p-th power turns out to respect addition as well as multiplication, so it is a ring homomorphism — the Frobenius endomorphism — and it is one of the defining features of characteristic-p algebra.

Precisely, let R be a commutative ring of prime characteristic p. The map F(x) = x^p is a ring endomorphism of R: it satisfies F(xy) = F(x)F(y) trivially and F(x + y) = F(x) + F(y) because the binomial coefficients C(p, k) for 0 < k < p are all divisible by p. On a field of characteristic p the kernel is trivial, so F is injective; on a finite field of order q = p^n the n-th iterate F^n is the identity.

Frobenius is the engine of positive-characteristic field theory. Surjectivity of F is exactly the condition that a field is perfect, its fixed field over F_p is F_p itself (giving Fermat's little theorem a^p = a), and on F_(p^n) the Frobenius generates the cyclic Galois group over F_p. In number theory its lifts to rings of integers control how primes split.