Theory of Fields

If F is a subfield of a field K, then K is said to be an extension of the field F. For $$\alpha \in K$$ , $$F(\alpha )$$ denotes the subfield generated by $$F\cup \{\alpha \}$$ , and the extension $$F\subseteq F(\alpha )$$ is called a simple extension of F. The element $$\alpha $$ is algebraic over F if $$\dim _FF(\alpha )$$ is finite. Field theory is largely a study of field extensions. A central theme of this chapter is the exposition of Galois theory, which concerns a correspondence between the poset of intermediate fields of a finite normal separable extension $$F\subseteq K$$ and the poset of subgroups of $$\textit{Gal}_F(K)$$ , the group of automorphis ms of K which leave the subfield F fixed element-wise. A pinnacle of this theory is the famous Galois criterion for the solvability of a polynomial equation by radicals. Important side issues include the existence of normal and separable closures, the fact that trace maps for separable extensions are non-zero (needed to show that rings of integral elements are Noetherian in Chap. 9 ), the structure of finite fields, the Chevalley-Warning theorem, as well as Luroth’s theorem and transcendence degree. Attached are two appendices that may be of interest. One gives an account of fields with valuations, while the other gives several proofs that finite division rings are fields. There are abundant exercises.