ABSTRACT
In the last chapter we obtained a good description for a simple extension of a field by an algebraic element. Specifically, if α is algebraic over F , we showed that F (α) is a vector space over F with dimension equal to the degree of α over F . Furthermore, every element of F (α) is algebraic over F . In this chapter we are interested in field extensions that are not necessarily simple, but in which every element is algebraic. We’ll then use our further results to provide more elegant proofs of the impossibility of two of the classical construction problems of the ancient Greeks.
