The Fundamental Theorem of Galois Theory

We saw in the last chapter that the Galois group of a finite extension of a field provides a lot of information about the structure of the extension field. In fact, if the extension is normal, then the degree of the extension is equal to the number of automorphisms belonging to the Galois group. In this chapter we encounter the Fundamental Theorem of Galois Theory, which shows that this connection between field extensions and groups carries even more information than that. In Chapter 48 we will be able to exploit this connection between field theory and group theory to address our goal of better understanding the solution of polynomial equations by field arithmetic and the extraction of roots.