3 problems

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

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 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 $q$…