These two are useful lemmas in Galois/Field Theory.
Finite extension is Algebraic extension (Proof)
Let be a finite field extension. Then is an algebraic extension.
Let be a finite extension, where . Let . Consider which has to be linearly dependent over since there are elements. Thus, there exists (not all zero) such that , so is algebraic over .
Finitely Generated Algebraic Extension is Finite (Proof)
Let be a finitely generated algebraic extension. Then is a finite extension.
Since is finitely generated, for some . Since is algebraic, each is algebraic over . Denote for . Then for each . Since is algebraic over , it is also algebraic over , so there exists a polynomial with coefficients in such that . Thus . Similarly . By Tower Law, .