Irreducible Polynomials and Finite Fields

This lecture covers irreducible polynomials with coefficients in a field, the Eisenstein criterion, quotients of polynomial rings, the order of the quotient field, properties of finite fields, the splitting field of a polynomial, and the cyclic group of units of a finite field. It also explains how to construct a unique finite field of p^n elements as a quotient of F_p[x] over the ideal generated by an irreducible polynomial of degree n.