Field $K$ with $\operatorname{Gal}(\overline{K}/K)\simeq\widehat{F_2}$

Question

Is there a field K such that $\operatorname{Gal}(\overline{K}/K)$ is the profinite free group with two generators?

For one generator I know that for all the $\mathbb{F}_p$ we have $\operatorname{Gal}(\overline{\mathbb{F}_p}/\mathbb{F}_p)\simeq\widehat{\mathbb{Z}}$.

Answer 1

Yes, there is. In fact, for every projective profinite group $G$, there is a perfect and pseudo-algebraically closed field whose absolute Galois group is $G$.

You may want to check corollary 23.1.2 of Field Arithmetic by Fried and Jarden for a proof.