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.

Proof:

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.

Proof:

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, .

Great post with a lot of detail.

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