51 problems

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…

A field $K$ is Hilbertian if it satisfies Hilbert's irreducibility property. The inverse Galois conjecture for Hilbertian fields. Over every Hilbertian field $K$, the inverse Galoi…

Assume that $E$ is a finite extension of $\mathbb Q_p$, and let $F$ be the function field of the schematic Fargues–Fontaine curve $X_C^{\mathrm{sch}}$ in the notation of the source…

Let $K$ be a finitely generated field over its prime field, and let Statements (a)--(c) in (3) of the theorem be the three assertions referenced in the source. Geyer–Jarden conject…

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…

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 $C_i$.

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…

Let $K$ be a field, let $p$ be a prime, and let $K^{(p)}$ be a maximal prime-to-$p$ extension of $K$. Write $K^{(p)}_{ab}$ for the maximal abelian extension of $K^{(p)}$. Bogomolov…

Krupiński's conjecture. Any rosy field is bounded and pseudo real closed (PRC).

Shapiro's field-independence conjecture. The admissibility of a fixed triple $[r,s,n]$ should be independent of the coefficient field, provided the characteristic is different from…

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

Junker–Koenigsmann conjecture. Every characteristic-zero field $K$ with finitely generated Galois group is ample.

Generalized Artin conjecture. If $K$ is a $C_d$-field, then

Generalized Artin conjecture. If $K$ is a $C_d$-field, then

Let $K$ be a field and let $A$ be a central simple algebra over $K$. Write $\operatorname{per}(A)$ and $\operatorname{ind}(A)$ for the period and index of $A$, respectively. A fiel…

NIP field conjecture. Every infinite NIP field is algebraically closed, real closed, or admits a non-trivial definable henselian valuation.

Let $K$ be a $C_d$-field for some integer $d>0$, and for each prime $p\in\mathbb{P}$ let $\operatorname{Brd}_p(K)$ denote the Brauer $p$-dimension of $K$. Artin's conjecture. The d…

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…

Large-field conjecture. The field ${{\mathbb Q^{\operatorname{ab}}}}$ is large.

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…

A field is bounded if it has only finitely many extensions of each given finite degree, and it is PAC if every geometrically irreducible variety over it has a rational point. Bound…

Let $\mathbb{Q}_{\mathrm{solv}}$ denote the maximal solvable extension of $\mathbb{Q}$. PAC conjecture. The field $\mathbb{Q}_{\mathrm{solv}}$ is pseudo-algebraically closed (PAC).…

Let $K$ be a field. A polynomial map $K\to K$ has cofinite image if its image has finite complement in $K$. Reineke's conjecture. If every non-constant polynomial map $K\to K$ has…