Proof that {1, pi} generates the extension R/Q

Question

I was trying to find a counterexample for: If E/K is a finitely generated extension then [E:K] must be finite

I know there is a proposition that states: E/K is a finitely generated extension and algebraic if and only if [E:K] is finite

So what i tried to do was find a non algebraic but finitely generated extension. Looking it up on the internet i found a claim that {1, pi} generates R over Q (and also [R:Q] is infinite). But I don't get how this can generate for example pi^2 since pi is transcendental to the rationals there is no way it could exists a,b in the rationals such that $\pi^2 = a * 1 + b * \pi$.

What I am missing?

PD: Yeah I know I should learn how to use Latex, but i do not, so sorry for the bad formatting. PD2: I didn't mention anywhere but obviously E and K are fields and K is contained in E (otherwise it wouldn't make any sense)

Answer 1

I assume that you mean finitely generated as a field extension. Then $K(T)$ (the field of rational functions) is a finitely generated field extension of $K$ which is never finite.

On the other hand, if $E/K$ is a field extension which is a finitely generated $K$-algebra, then it is finite (Zariski's Lemma).

Of course, the claim about $\pi$ is wrong.