3 problems

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, let $r>1$ be an integer, let $p\ne2$ be a prime, and let $S$ contain…

Let $L/K$ be a finite Galois extension of number fields with arbitrary Galois group $G$, let $r>1$ be an integer, and let $p$ be an odd prime. Let $\mathcal J_r^S$ be the canonical…

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, let $r>1$ be an integer, and let $p$ be an odd prime. Let $\mathcal J_r^S$ be the canonical fractiona…