Polynomials

Abstract

Let F be an arbitrary field. We will study the polynomial ring F[x] in this chapter. We will consider what it means for one polynomial to divide another, and we will formalize the process of long division. We will consider the notion of irreducibility of polynomials, and we will see that F[x] behaves remarkably like the integers when it comes to factorizations of polynomials, with irreducible polynomials playing the role of prime numbers. In fact, we will prove that every polynomial factors into a product of irreducible polynomials, and that the irreducible polynomials that occur in this factorization are unique in a suitable sense! We will end with a discussion on the number of roots of a given polynomial.