253 problems

Let $p$ be an odd prime and let $t$ be a positive integer. A number field $K$ is $p$-rational if, for the set $S$ of primes of $K$ lying above $p$, the Galois group of the maximal…

Gras's conjecture. For $p\gg 0$, one has

Let $A$ be a random reciprocal polynomial in the restricted coefficient model, with coefficients chosen independently and uniformly from … and condition on $A(0)=1$. Odlyzko–Poonen…

Efrat's arithmetical Elementary Type Conjecture. There is such a decomposition where, for each $1\leq i\leq n$, one of the following holds: $G_i\cong\mathbb{Z}_p$; $G_i$ is the dec…

Positselski's conjecture. Absolute Galois groups of fields containing a $p$-th root of unity satisfy the Koszul property at $p$.

Let $K$ be a global field, and let $K^{\mathrm{cyc}}$ denote its maximal cyclotomic extension. The absolute Galois group of a field $E$ is denoted by $G_E$. Shafarevich's freeness…

Kummer-faithfulness conjecture. Any finite extension of $\overline{K}(\sigma)$ is Kummer-faithful for almost all $\sigma\in G_K^e$.

Let $E/K$ be an elliptic curve over a number field $K$, and let $G_K:=\operatorname{Gal}(\overline{K}/K)$. Let $G$ be a topologically finitely generated subgroup of $G_K$, and writ…

Let $p$ be an odd prime, let $K$ be a number field, and let $S$ be a finite set of tame primes of $K$. For a set $T$ of primes disjoint from $S$, let $G_{K,S}^T$ denote the Galois…

Let $G$ be a nontrivial finite group, and let $r(G)$ denote the least number of places of ${\mathbb Q}$ ramified in a finite Galois extension $K/{\mathbb Q}$ with…

Let $(F_n^{(k)})$ be the $k$-generalized Fibonacci sequence, and let $G$ be the Galois group associated with its characteristic polynomial, viewed as a subgroup of the symmetric gr…

Let $T$ be the rooted tree associated with iterated preimages of a rational function of degree two, and let an arboreal representation be the continuous action on this tree arising…

Let $n\geq 2$, let $Q_n$ be the polynomial studied in the paper, and write … For a polynomial with rational coefficients, let $\operatorname{Gal}(Q_n)$ denote its Galois group and…

Gross's conjecture. For every prime $p$, there exists a non-solvable number field $K/{\mathbf Q}$ ramified only at $p$ and possibly at $\infty$.

Kernel Unipotent Conjecture. The Zassenhaus filtration of $G_K$ is given by the kernels of unipotent representations.

Let $x_1,\dots,x_n$ be Galois conjugates, and consider a relation $\sum a_i x_i=0$ with coefficients $a_i$. The relevant local conditions are the necessary conditions established b…

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 $f_b(z)=z^{3}+b$ be a unicritical cubic polynomial, and fix integers $k\geq 0$ and $n\geq 1$. Consider the finite set of complex values of $b$ for which the critical point $0$…

Let $\sigma=[n]$ with $\alpha_1=0$, and let ${\cal L}_{[n]}$ be the canonical LSSM whose parameter space consists of the symmetric matrices … Let $G_{\mathcal{L}_{[n]}}$ denote the…

Let $p\ge 5$ be an odd prime, and let … where $a=\sigma\cdot prb$, $\sigma\in\lbrace -1,+1\rbrace$, $r\in\mathbb{N}$, and $b\in\mathbb{N}$ is squarefree. Let $N$ be the splitting f…

Let $p$ be a prime. A pro-$p$ group is of elementary type if it belongs to the class generated from $\mathbb{Z}_p$ and Demushkin groups by the constructions specified in the source…

Positselski's pro-$\ell$ Bogomolov conjecture. The maximal pro-$\ell$ Galois group of $\sqrt[\ell^\infty]{\mathbb{K}}$ is a free pro-$\ell$ group.

Let $K$ be a finitely generated field over its prime field, let $e$ be a positive integer, and let $\sigma$ range over $\mathrm{Gal}(K^\mathrm{sep}/K)^e$. For almost all such…

A pro-$2$ Galois group is the Galois group of the maximal pro-$2$ extension of a field; a pro-$2$ group is ETG$_2$ if it can be constructed from $\mathbbm{Z}/2\mathbbm{Z}$, infinit…

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…