4 problems

Let $K\subset\overline{\mathbb Q}$ and suppose that the absolute Galois group $G_K$ is a projective profinite group. The group $G_K$ is profree if it is a free profinite group. Hil…

Bottom theorem. For almost all $\boldsymbol{\sigma}\in\operatorname{Gal}(K)^e$ and every proper subfield $N\subsetneqq K_s(\boldsymbol{\sigma})$, if the extension…

Let $K$ be a fully Hilbertian field, and let $G_K$ denote its absolute Galois group. A profinite group of infinite rank $m$ is semi-free if every finite split embedding problem for…

Bottom theorem conjecture. For almost all \text{\boldmathsigma}\in\operatorname{Gal}(K)^e, the field K_s(\text{\boldmathsigma}) is a finite separable extension of no proper…