10 problems

Jarden's conjecture. Every such intermediate field $L$ is Hilbertian. The source states that this conjecture was established by a series of works, with extensions to commutative al…

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 $K$ be a hilbertian field. A split finite embedding problem over $K$ is a finite embedding problem for the absolute Galois group of $K$ whose corresponding finite group extensi…

A split finite embedding problem (FSEP) over a field $K$ consists of a finite Galois extension and a split surjection of finite groups associated with it; it is solvable if there i…

Let $K$ be a Hilbertian field. A finite split embedding problem over a field $E$ consists of a finite Galois extension $F_1/E$ and a split epimorphism of finite groups … A solution…

Let $K$ be a Hilbertian field, let $A$ be an abelian variety defined over $K$, and let $M$ be an intermediate field of $K(A_{{\mathrm{tor}}})/K$. Kuykian conjecture. The field $M$…

Let $K$ be a Hilbertian field, meaning that every irreducible polynomial $f(T,X)$ that is separable in $X$ admits a specialization $a\in K$ such that $f(a,X)$ remains irreducible.…

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…