Splitting Fields

Abstract

We consider field extensions which are generated by all zeros of a given polynomial over a field containing its coefficients. Such a field is called a splitting field. Splitting fields of polynomials play a central role in Galois theory. It is intuitively clear that a splitting field of a polynomial is a very natural object carrying a lot of information about it. We will find a confirmation of this in the following chapters, in particular, when we come to the main theorems of Galois theory (Chap. 9) and its applications in subsequent chapters (e.g. in Chap. 13, where we study the solvability of equations by radicals). As an important example, we study finite fields in this chapter. These can be easily described as splitting fields of very simple polynomials over finite prime fields. We further consider the notion of an algebraic closure of a field K, which is a minimal field extension of K, containing a splitting field of every polynomial with coefficients in K.