The Content of a Polynomial (Algebra 2 Lecture 9 Video 2)

Lecture 9: We started this lecture by proving that if R[x] is a UFD then R is a UFD. The main goal of the rest of this lecture was to prove the converse, that if R is a UFD then R[x] is a UFD. Let R be a UFD and F be its field of fractions. We proved Gauss' Lemma, which helps to explain the relationship between factorizations of a polynomial p(x) in R[x] and factorizations in F[x]. We defined the content of a polynomial in R[x] and proved that content is multiplicative. We then proved the main result of this lecture. At the end of the lecture we talked briefly about factorizations of monic polynomials in R[x] where R is an integral domain that is not necessarily a UFD.