Galois theory: Frobenius automorphism

This lecture is part of an online graduate course on Galois theory. The presenter introduces the Frobenius endomorphism in characteristic p, noting that for finite fields it is an automorphism generating the Galois group. He then addresses the challenge of defining a Frobenius automorphism in characteristic zero by reducing an integral form of a number field modulo a prime. Assuming the polynomial defining the field is monic with integer coefficients and separable modulo p, he constructs a Frobenius automorphism that acts on the reduction. He proves that this automorphism lifts to an element of the Galois group of the original field. As an example, he applies this to the Gaussian integers, showing that the Frobenius automorphism maps i to i^p, and uses this to prove that -1 is a square modulo an odd prime p if and only if p ≡ 1 mod 4. The lecture concludes with a preview of cyclotomic polynomials.