6 problems

Hilbertian-projective implies pro-free conjecture. If $K$ is Hilbertian and projective, then $G_K$ is profree.

Mináč–Tăn's conjecture. For every field $F$, every prime $p$, every $n\geq 3$, and all $\chi_1,\dots,\chi_n\in H^1(F,\mathbb Z/p\mathbb Z)$, if the Massey product

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…

Let $F$ be an infinite field with finitely generated absolute Galois group. Koenigsmann's conjecture. Then $F$ is ample. This conjecture concerns which fields are ample and is pres…

Let $\mathbb{Q}_{\rm ab}$ denote the maximal abelian extension of $\mathbb{Q}$, and let $G_{\mathbb{Q}_{\rm ab}}$ be its absolute Galois group. Shafarevich's conjecture. The group…

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…