2 problems

Let $\mathbb{K}$ be a Galois extension of $\mathbb{Q}$ of degree $n$, and let $p$ be a prime such that $p\mid\mathfrak{J}(\mathbb{K}\mid\mathbb{Q})$. Let $\rho(p)$ denote the numbe…

Let $S=R=\mathbb{Z}[i]$, and let $(l_1,l_2,\ldots)$ be the sequence of lengths associated with a fixed divisor sequence, where $l_k$ is the smallest number of initial elements dete…