7 problems

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

Positselski–Quadrelli conjecture. The subgroup $K(\mathcal{G}_{\mathbb{K},p})$ is a free pro-$p$ group for every field $\mathbb{K}$ containing $\mu_p$.

Let $p>3$ be prime. For a prime $q\equiv3\pmod4$, write $f$ for the order of $q$ modulo $p$ and set $\kappa=(q^f-1)/p$. Let $\xi=e^{2\pi i/n}$ for each integer $n>2$ dividing $q-1$…

Let $p>3$ be prime. For a triple $(p,u,v)\in\mathbb Z^3$ with $u$ and $v$ coprime and $p\mid v$, let $q$ be a $p$-principal prime with $puv\not\equiv0\pmod q$, let $n$ be the order…

Let $p>3$ be prime. For a $p$-principal prime $q\equiv3\pmod 4$, let $f$ be the order of $q$ modulo $p$ and set $\kappa=(q^f-1)/p$. Furtwängler-type Kummer criterion. There exists…

Let $p>3$ be prime and let $(p,u,v)\in\mathbb Z^3$ with $u$ and $v$ coprime and $p\mid v$. For a $p$-principal prime $q$ with $q\nmid puv$, let $n$ be the order of $v/u$ modulo $q$…

Motivic extension conjecture. If $\mathcal{R}$ is replaced by a suitable category of motives, this is the only way to construct such extensions.