3 problems

Let $p$ be an odd prime, let $K=\mathbb{Q}(\zeta)$ with $O_K=\mathbb{Z}[\zeta]$, and let $A$ be the matrix defined by … for $\beta=\sum_{i=0}^{p-2}b_i\zeta^i\in K$. Let $D$ be a…

The determinant-$p$ conjecture. The matrix $A=[a_{ij}]$ is a $(p-1)\times(p-1)$ matrix with determinant $p$.

Let $K$ be a number field, let $g$ be the number of primes of $K$ above a rational prime, and let $(\mathbf{e},\mathbf{f})\in\mathbb{N}^g\times\mathbb{N}^g$ satisfy … Here…