Galois Theory of Finite Fields: Some Applications (Algebra 3: Lecture 22 Video 2)

Lecture 22: We started this lecture by reviewing some things that we know about extensions of finite fields.  We have seen that subfields of F_{p^n} are in bijection with divisors d of n.  We saw that each of these subfields gives a Galois extension of Fp that is cyclic and we described a generator for it.  After talking about finite extensions of Fp we mentioned that there is an analogous set of results for finite extensions of any finite field (not necessarily of prime order).  We discussed a neat application-- x^4+1 is irreducible in Z[x], but is reducible modulo p for each prime p.  We proved that F_{p^n} is a simple extension of Fp and gave the number of primitive elements for this extension.  We proved that x^{p^n}-x is the product of all the distinct irreducible polynomials of degree d in Fp[x] where d runs through all divisors of n.  We saw how this led to a formula for the number of irreducibles in Fp[x] of degree 1,2, and 4.  We mentioned that using the Möbius Inversion Formula, one can prove a general formula for the number of irreducibles of degree n. At the end of the lecture we discussed isomorphisms between finite fields.