4 problems

Let $K'$ be the relevant extension, let $H'$ be its Galois group, let $I(H')$ be the augmentation ideal, and let $r_{K'}$ be the rank of the $S(K')$-unit group. Let $\theta_{H'}$ be…

Let $L/K$ be the Galois extension and $G=\operatorname{Gal}(L/K)$, with $G^+=G/\langle\tau\rangle$, and let $e_r$ denote the relevant idempotent. Let $L_{p,S}(r)$ be the central el…

Stark's conjecture over $\mathbb{Z}$. There is a unique element

Suppose $F$ is a real quadratic field and $K_n=F(\boldsymbol{\mu}_n)$, where $n$ is prime to the conductor of $F/\mathbf{Q}$. Let…