24 problems

Let $K$ be an infinite field, let $X$ be a curve over $K$, let $K^{\operatorname{sep}}$ be a separable closure of $K$, and let $G_K$ be the absolute Galois group of $K$. For…

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

Let $K$ be a $C_d$ field, meaning that every homogeneous form of degree $e$ in more than $e^d$ variables over $K$ has a nontrivial zero, and let $\alpha\in\operatorname{Br}(K)$. Th…

Hyperbolic polynomial branch and extension conjecture. The following are equivalent:

Kato–Kuzumaki's conjecture. A field $k$ satisfies $C_i^q$ if and only if

Let $K$ be an infinite field. It is model-theoretically minimal if every definable subset of $K$ is finite or cofinite. Podewski's conjecture. If $K$ is model-theoretically minimal…

Period-index conjecture. For every $\alpha \in \operatorname{Br}(K)$,

Let $K\leq\bar{\mathbb Q}$ be Hilbertian, meaning that Hilbert's irreducibility theorem holds in $K$, and let $G_K$ be its absolute Galois group. Assume that $G_K$ is projective am…

Let $k$ be a Hilbertian field. A finite split embedding problem over $k$ is an epimorphism … where $G$ is finite, $L/k$ is Galois, and there is an embedding…

Stable-field and simple-field conjectures.

Let $G$ be a group, and let $G-\mathrm{TCF}$ denote the theory of existentially closed $G$-fields, when such a theory exists. Hrushovski's nonexistence conjecture. If $G$ has a sub…

Simple-field reduction conjecture. 1. $K$ is strongly IAC; and 2. If $K$ is VAC, then $K$ is pseudo-algebraically closed.

Simple-field conjecture. Every simple field is pseudo-algebraically closed.

Let $K$ be a field finitely generated over $\mathbb{Q}$, let $X$ be a pointed nonsingular curve over $K$, and let $\bar K(\boldsymbol{\sigma})$ denote a field obtained by fixing a…

A field $K$ is ample if every nonsingular curve over $K$ has either no $K$-points or infinitely many. Junker–Koenigsmann's conjecture. Every infinite field with finitely generated…

Lang's conjecture. The function field $\mathbb R(X)$ is $C_i$.

For a field $F$, define property $(P)$ by requiring that every polynomial $g(x)\in F[x]\setminus F$ for which there is an infinite set $X\subseteq F$ satisfying $g(X)=X$ has degree…

Let $K$) be a field that is not locally finite, and let … be topologically finitely generated. Let $E/K$ be a non-trivial elliptic curve. Infinite-rank conjecture. The elliptic cur…

Let $F$ be a field and let $(\operatorname{Br}F)_p$ denote the $p$-primary component of its Brauer group. Brumer–Rosen conjecture. For each prime $p$, one of the following holds: 1…

Let $\mathbf{Q}^{\mathrm{ab}}$ denote the maximal abelian extension of the rational numbers. Artin's conjecture. The field $\mathbf{Q}^{\mathrm{ab}}$ is $(C_1)$. The source present…

Let the degree problem ask whether equivalent number fields must have the same degree. Degree-problem conjecture. The degree problem has a negative answer. Moreover, there exist a…

Let $G$ be a finite group. A group is Sylow metacyclic if all of its Sylow subgroups are metacyclic. Schacher's conjecture. The group $G$ is $Q$-admissible if and only if $G$ is Sy…

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…