Cyclotomic Polynomials and Cyclotomic Extensions (Algebra 3: Lecture 15 Video 2)

Lecture 15: We started this lecture by proving that for any prime p and positive integer n, there exists a finite field of order p^n. We then proved that this field is unique up to isomorphism, that any finite field of order p^n is isomorphic to it.  We then talked about how this construction related to some of our earlier discussions about finite fields.  The rest of this lecture was focused on proving that the degree of the nth cyclotomic field over Q is phi(n).  We defined the nth cyclotomic polynomial and gave a sketch of the argument.  Our goal is to show that this polynomial is the minimal polynomial of a primitive nth root of unity over Q.  We discussed how to compute the nth cyclotomic polynomial and some questions about its coefficients.  We then proved that cyclotomic polynomials have integer coefficients.  In the last video, we proved that the nth cyclotomic polynomial is irreducible.