100 problems

Let $n,m\in\mathbb{Z}$ and $A,B\in\mathbb{Z}\setminus\{0\}$ satisfy $1\le m<n$ and $\gcd(n,mB)=1$. Suppose that $f(X)=X^n+A(BX+1)^m\in\mathbb{Z}[X]$ is irreducible, let $\alpha$ be…

For every prime $p$ and every integer $r\ge 1$, there exist infinitely many integers $D>0$ such that the fields…

For every totally real number field $k$ and every prime number $p$, let $k_\infty^{\mathrm{c}}/k$ be the cyclotomic $\mathbb{Z}_p$-extension, and let $M(k_\infty^{\mathrm{c}})$ be…

Let $T^n$ be the $n$-torus, let $\phi$ be a quasiperiodic flow on $T^n$ generated by a constant vector field $X$, and let $F$ be a real algebraic number field of degree $n$ over…

For integers $n>2$ and $k>1$, let $F_{n,k}(x)=\Phi_n(1+x+\cdots+x^k)\in\mathbb{Q}[x]$, where $\Phi_n$ is the $n$-th cyclotomic polynomial. Determine, exactly in terms of $n$ and…

Brumer's conjecture.

Oppenheim's conjecture. The lattice

Let $q$ be a prime number, and let $h_1(q)$ denote the relative class number. Define … The ratio $R(q)$ is called the Kummer ratio. Kummer's conjecture. As $q$ tends to infinity, ……

Let $L/K$ be a finite weakly ramified Galois extension of number fields, with $G=\operatorname{Gal}(L/K)$, ring of integers $\mathcal{O}_L$, and different $\mathcal{D}_{L/K}$. When…

Let $d=n^2+3=p$ be a prime in the specified sequence, let $K$ be the real quadratic field used in the construction, and let $e^{i\vartheta_j}$ be the phases in the almost-flat fidu…

Stark's conjecture over the rationals. For every $\chi\in\widehat{G}$, one has $A(\chi,\mathcal{A})\in\overline{\mathbb{Q}}$ and

Let $\varphi=\frac{1+\sqrt{5}}{2}$ be the golden ratio, and let $\langle\varphi\rangle=\{k\varphi:k\in\mathbb{Z}\}$ be the subgroup of $\mathbb{C}$ generated by $\varphi$. An irred…

Rational Stark conjecture. There exists an element $u_\chi\in(k\otimes_{\mathbb{Z}}U_{v,S})^{\chi^{-1}}$ such that

Fix an integer $d \geq 3$. For a totally real number field $K$ of degree $d$, let $\operatorname{disc}(K)$ denote its discriminant, and let an ideal class mean a class of fractiona…

Let $N$ be a prime level, let $\chi$ be the character used to define $h_\chi$, and let $B_{m,\chi}$ denote the $m$-th generalised Bernoulli number attached to $\chi$. Write … For e…

Algebraicity–multiplier group conjecture. A quasiperiodic flow $\psi$ is $F$-algebraic if and only if $M_\psi$ is a finite index subgroup of ${\mathfrak o}_F^*$.

Let $n\geqslant 3$ and let $\Lambda\subset\mathbb{R}^n$ be an $n$-dimensional lattice with positive norm minimum $N(\Lambda)>0$. A lattice is called algebraic when it is similar, m…

Let $p$ be an odd prime splitting completely in $k$, let $K/k$ and the other notation be as in the paper, and let $G=\operatorname{Gal}(K/k)$. Let $\eta^+_{K/k}$ be the image of th…

Let $K^+/k$ be the totally real subextension under consideration, let $G^+=\operatorname{Gal}(K^+/k)$, let $S_0=S_{\rm ram}(K/k)$, and let $U_{S_0}(K^+)$ be the group of global…

Let $k$ be a totally real number field, let $p$ be an odd prime splitting completely in $k$, and let $K$ be as in the paper. Write $G=\operatorname{Gal}(K/k)$ and let $\delta(K/k)$…

Let $\phi$ be a quasiperiodic flow on $T^n$ generated by a constant vector field $X$. Call $\phi$ algebraic if it is $F$-algebraic for a real algebraic number field $F$ of degree…

Let $F$ be a real algebraic number field of degree $n$ over $\mathbb{Q}$, and let $T^n$ denote the $n$-torus. The multiplier-group equality conjecture. There exists a quasiperiodic…

Let a SIC-POVM in $\mathbb{C}^d$ have normalized overlaps $e^{i\theta_{jl}}$, where the angles $\theta_{jl}$ are defined by the normalized inner products of its fiducial vectors. A…

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

Let $p$ be the odd prime and let $\theta$ be the algebraic number considered in the preceding discriminant computation, with $\mathbb{Q}(\theta)$ its generated number field. Discri…