The Multiplicative Group of a Finite Field is Cyclic (Algebra 2: Lecture 11 Video 2)

Lecture 11: We started this lecture by discussing quotients of F[x]. We saw that the maximal ideals in F[x] are exactly the ideals (f(x)) generated by irreducible polynomials f(x). We then saw how to understand the structure of F[x]/(g(x)) in terms of the factorization of g(x). This was an application of the Chinese Remainder Theorem. We then built on a result from the previous lecture to show that a polynomial of degree d in F[x] can have at most d roots in F[x], even when roots are counted with multiplicity. We used this to prove that a finite subgroup of the multiplicative group of a field is cyclic. We gave two proofs of this result. We then showed how to use this result to understand the structure of the group of units of Z/nZ.