4 problems

Let $K=\mathbb{Q}(\sqrt[8]{5})$ be the pure octic field generated by an eighth root of $5$. The minimal index conjecture asserts that the minimal index of $K$, namely the least ind…

Let $r\in\{1,7,18,24\}$, let $m=r+25k$ be square-free, and let $K=\mathbb{Q}(\sqrt[5]{m})$. The minimal index of $K$ is the least index of an algebraic integer generating a power i…

Let $K=\mathbb{Q}(\sqrt[4]{m})$ with $m=9+16k$ square-free. Monogenity conjecture. The only monogenic fields in this family are … The preceding discussion gives these two examples…

Let $K=\mathbb{Q}(\sqrt[4]{m})$ be a pure quartic field with $m=9+16k$ square-free. The minimal index of $K$ is the least index of an algebraic integer generating a power integral…