Galois theory: Finite fields
Channel: Richard E Borcherds
Date: 2020-12-30

This lecture is part of an online graduate course on Galois theory.

We use the theory of splitting fields to classify finite fields: there is one of each prime power order (up to isomorphism).

We give a few examples of small order, and point out that there seems to be no good choice for a standard finite field of given order: this depends on the choice of an irreducible polynomial.

Finally we show how to count the number of irreducible polynoials of given degree in a finite field (using the field of order 64 as an example).

The lecture mentioned that there does not seem to be a clear choice for the  "best" of the three irreducible polynomials of degree 4 over F2. Here is a poll where viewers can vote on which they prefer and see how others have voted:
https://forms.gle/EsTzdTkFd6k9RsAW8