Definition 2

— An extension E of a field K is said to be finitely generated if it has a finite generating family. It is called monogenous if there exists x in E such that $E = K(x)$.

The Cor. of Prop. 3 shows that every extension E of a field K is a directed union of finitely generated extensions contained in E. It is clear that every extension E of K of finite degree is also finitely generated because a basis of E (considered as vector space over K) is also a generating family of E over K; we shall see later that the converse does not hold.