Take $K=\mathbb Q$. For every positive <integer> $n$, the real number $\sqrt[n]2$ lies in $\overline{\mathbb Q}$ and has degree $n$, by <Eisenstein's criterion> at $2$. If $\overline{\mathbb Q}/\mathbb Q$ had finite degree, all its subextension degrees would be bounded by that degree, contradicting this unbounded family. Therefore \b[$[\overline{\mathbb Q}:\mathbb Q]$ is infinite].

Solution

Past exam of the number theory course of the University of Oxford / 2010 / A / 3 / ii / b / Solution

 Solution (source code)