Cyclotomic Swan subgroups and primitive roots

Abstract

Let $K_m=\mathbb{Q}({\zeta }_m)$ where ${\zeta }_m$ is a primitive $m$th root of unity. Let $p>2$ be prime and let $C_p$ denote the group of order $p.$ The ring of algebraic integers of $K_m$ is $\Cal{O}_{m}=\Bbb{Z}[ζ_{m}].$ Let ${\Lambda }_{m,p}$ denote the order $\Cal{O}_{m}[C_{p}]$ in the algebra $K_m[C_p].$ Consider the kernel group $D({\Lambda }_{m,p})$ and the Swan subgroup $T({\Lambda }_{m,p}).$ If $(p,m)=1$ these two subgroups of the class group coincide. Restricting to when there is a rational prime $p$ that is prime in $\Cal{O}_{m}$ requires $m=4$ or $q^n$ where $q>2$ is prime. For each such $m$, $3\le m\le 100,$ we give such a prime, and show that one may compute $T({\Lambda }_{m,p})$ as a quotient of the group of units of a finite field. When $h_{mp}^{+}=1$ we give exact values for $|T({\Lambda }_{m,p})|$, and for other cases we provide an upper bound. We explore the Galois module theoretic implications of these results.