It's well known that given a field $K$, an extension $F/K$ is finite if and only if $F$ is algebraic and finitely generated. Naturally, this suggests the existence of algebraic extensions which are not finitely generated. In trying to think of such examples, the only one that I came up with would be the algebraic closure of $\mathbb{Q}$.
Are there examples of non-finitely generated algebraic extensions which can be described without saying it's the algebraic closure of something? Mainly because the existence of the closure doesn't give me a good intuition as to exactly what the extension field is composed of.
Best Answer
Not sure what description you want to see. For example, $F=\mathbb{Q}(\{\sqrt[n]{2}: n\in\mathbb{N}\})$ is clearly an algebraic extension of $\mathbb{Q}$ which is not finite, and so not finitely generated.
A short explanation to why the extension is infinite: for each $n$ the polynomial $x^n-2\in\mathbb{Q}[x]$ is irreducible by Eisenstein's criterion, and so $\sqrt[n]{2}$ is an algebraic number of degree $n$ over $\mathbb{Q}$. So we have:
$[F:\mathbb{Q}]\geq [\mathbb{Q}(\sqrt[n]{2}):\mathbb{Q}]=n \ \ \forall n\in\mathbb{N}$
And so the extension degree is infinite.