Why the field extension $\mathbb{C}/\mathbb{Q}$ is not finitely generated?

Question

I know the extension $\mathbb{C}/\mathbb{Q}$ is not finite, as $\mathbb{Q}$ is countable whereas $\mathbb{C}$ is not.

However, how should I prove that the field extension $\mathbb{C}/\mathbb{Q}$ is not finitely generated?

Answer 1

We just need to show that if $K$ is a field with countable (infinite) cardinality, then any countably generated field $L$ over $K$ is also countable. By induction, it's sufficient to show any simple extension $K(\alpha)$ is countable. And indeed $$K(\alpha)=\cup_{n=1}^\infty \{ \frac{\sum_{i=0}^n a_i\alpha^i}{\sum_{i=0}^n b_i\alpha^i} | a_i, b_i\in K, \sum_{i=0}^n b_i\alpha^i\not=0\}$$

Note that the cardinality of $\{ \frac{\sum_{i=0}^n a_i\alpha^i}{\sum_{i=0}^n b_i\alpha^i} | a_i, b_i\in K, \sum_{i=0}^n b_i\alpha^i\not=0\}$ won't exceed $|K|^{2n}$, hence must be countable as well.

So just to recognize there aren't so many polynomials (or fractions), which was first known to Cantor who proved there exist transcendental numbers this way without exhibiting a single example.