Can an algebraic extension of an uncountable field be of uncountable degree?

Question

I want to show that, if $K$ is a maximal subfield of $\mathbb C$ without $\sqrt{2}$ in it, then $\mathbb C$ is of countable degree over $K$. Is it the case that an algebraic extension of an uncountable field always has countable degree? if so, I can try to show $K$ is uncountable.

Answer 1

Yes. Here's an example.

Suppose that $B$ is an uncountable set of algebraically independent (over $\Bbb Q$) numbers from $\Bbb C$, and consider $F$ to be the smallest field containing $B$ and $\Bbb Q$.

Now consider $K=F[\{\sqrt x\mid x\in B\}]$. Since the elements of $B$ are algebraically independent it follows that none of the square roots is in $F$. But this is certainly an algebraic extension.