6 problems

Let $K_a=\mathbb{Q}(\rho_a)$ be a Shanks simplest cubic field, where $\rho_a$ is a root of $f_a(x)=x^3-ax^2-(a+3)x-1$, and let $\varepsilon,\delta\in\mathbb{Z}[\rho_a]^\times$ sati…

Let $K_a$ be the simplest cubic field considered in the paper, let $\mathcal{O}_{K_a}$ be its maximal order, and let $u_1,u_2\in\mathcal{O}_{K_a}^*$ and $n\in\mathbb{Z}$ satisfy th…

Let $K_a$ be the simplest cubic field considered in the paper, let $\rho$ generate the order $\mathbb{Z}[\rho]$, and let $u_1,u_2\in\mathbb{Z}[\rho]^*$ and $n\in\mathbb{Z}$. Two so…

Let $A$ be a finite-dimensional division algebra over $\mathbb{Q}$, and let $\Gamma_1,\Gamma_2$ be finitely generated semigroups of the multiplicative group $A^\times$. Fix…

Conjecture. For any $k\geq 3$ there exists a number field $K$ with $\ell(K)=k$.

Let $p>0$ be a prime, let $K$ be a field of characteristic $p$, and let $G$ be a subgroup of $K^*\times K^*$ with finite rank … For given $a,b\in K^*$, consider the equation … in…