Unsolved Problems

Is every finite group the Galois group of some Galois extension of the rational numbers $\mathbb{Q}$?...

A set of conjectures about group rings: (1) Zero divisor conjecture: If $G$ is a torsion-free group and $K$ is a field, then $K[G]$ has no zero diviso...

A ring has no non-zero nil ideal (an ideal all of whose elements are nilpotent) if and only if it has no non-zero nil one-sided ideal....

If $F: \mathbb{C}^n \to \mathbb{C}^n$ is a polynomial map with constant non-zero Jacobian determinant, then $F$ is invertible....

Prove that the general equation of the seventh degree cannot be solved using functions of only two variables....

Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?...

Can every non-negative rational function be expressed as a sum of squares of rational functions?...

What is the largest product-free set in the alternating group $A_n$?...

Which finite groups have the smallest largest product-free sets?...

Do $\Phi(G)$ and $\Phi'(G)$ coincide?...

Is every group well-approximated by finite groups?...

Suppose $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \subset A$ with $|S| \gg K^{-O(1)}|A|$ and $S^8 \subset A^4$?...

Pick $x_1, \dots, x_k \in A_n$ at random. Is it true that, almost surely as $n \to \infty$, the random walk on this set of generators and their invers...

Find bounds in the classification theorem for approximate groups....

Is every group well-approximated by finite groups?...

For every positive integer $k$, does there exist a Hadamard matrix of order $4k$?...

If a ring has no nil ideal other than $\{0\}$, does it follow that it has no nil one-sided ideal other than $\{0\}$?...

Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?...

For a left-and-right Noetherian ring $R$, is the intersection of all powers of the Jacobson radical $J(R)$ equal to zero?...

Do SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite dimensions?...

If a univariate polynomial $f$ of degree $d$ over a field of characteristic 0 shares a common factor with each of its first $d-1$ derivatives, must $f...

Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugati...

For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2,5)$ finite?...

If a finite system of left cosets of subgroups of a group $G$ partitions $G$, then must at least two of the subgroups have the same index in $G$?...

Does there exist a rectangular cuboid where all edges, face diagonals, and space diagonals have integer lengths?...

For a matroid of rank $n$ with $n$ disjoint bases $B_1, \ldots, B_n$, can we always find an $n \times n$ matrix whose rows are the bases and whose col...

For a finite group $G$ and prime $p$, is the number of irreducible complex characters of $G$ whose degree is not divisible by $p$ equal to the corresp...

Is every group surjunctive? That is, for any group $G$, if $\phi: A^G \to A^G$ is a cellular automaton that is injective, must it also be surjective?...

Can graph isomorphism be decided in quasi-polynomial time for all graphs?...

Does the order of the center of the Steinberg group of the ring of integers of a number field relate to the value of the Dedekind zeta function at $s=...

Can Schubert calculus be given a rigorous foundation?...

What is the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$ in the plane?...

Is there a bound $B(g, d)$ such that every curve of genus $g$ over a number field of degree $d$ has at most $B(g, d)$ rational points?...

Is every piecewise-polynomial function $f: \mathbb{R}^n \to \mathbb{R}$ the maximum of finitely many minimums of finite collections of polynomials?...

If $R$ is a regular local ring and $P, Q$ are prime ideals with intersecting dimensions satisfying a certain condition, is the intersection multiplici...

Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugati...

For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2, 5)$ finite?...

What are the composition factors of finite groups appearing in genus-0 systems?...

If a finite system of left cosets of subgroups of a group $G$ partitions $G$, must some two subgroups have the same index?...

Is every finite group the Galois group of some Galois extension of $\mathbb{Q}$?...

Is there an algorithm to determine whether two Coxeter groups given by presentations are isomorphic?...

Are there infinitely many Leinster groups?...

Does generalized moonshine exist for all elements of the Monster group?...

Is every finitely presented periodic group finite?...

Is every group surjunctive?...

Is every discrete countable group sofic?...

What is the structure of the discrete spectrum of automorphic forms on reductive groups?...

Is there a relationship between the numbers of irreducible characters in blocks of a finite group and its local subgroups?...

Can representations of semisimple algebraic groups be characterized over the integers?...

How do values of Kazhdan-Lusztig polynomials at $1$ relate to multiplicities of irreducible representations in Verma modules?...