2 problems

Let $p$ be an odd prime, let $F_\infty=\mathbb{Q}(\mu_{p^\infty},\sqrt[p^n]{m}:n\geq1)$, let $G=\operatorname{Gal}(F_\infty/\mathbb{Q})$, and let…

Let $E/bbQ$ be an elliptic curve with good ordinary reduction at an odd prime $p$, and let $F_\infty=\bigcup_n\mathbb{Q}(\mu_{p^n},\sqrt[p^n]{m})$. For a self-dual Artin representa…