finitely generated and algebraic and a finite extension

Question

Let $L/K$ be an extension.

I have proven that $L/K$ is finite if and only if $L/K$ if finitely generated and algebraic.

I have found a proof online which used the following theorem:

$\alpha$ is algebraic over $K$ if and only if $K(\alpha)/K$ is finite.

could someone perhaps tell me the link from what I've proven to the theorem I have found online?

i.e I want to use the theorem: $L/K$ is finite if and only if $L/K$ if finitely generated and algebraic.

to prove: $\alpha$ is algebraic over $K$ if and only if $K(\alpha)/K$ is finite.

Answer 1

Well, we can use our definition of an algebraic number to get one of the directions pretty quickly. Recall that a number is algebraic if it is the root of a polynomial expression, that is, $\alpha$ is a root of some polynomial $f(x)=a_nx^n+ \dots a_1x+ a_0$. Since the degree of an extension is the degree of the minimal polynomial, we know that the degree of $\alpha$ must be finite (and here, the degree must be $\leq n$) if it is algebraic. The other direction follows similar ideas.