### Cyclotomic Swan subgroups and Primitive Roots, by Timothy Kohl and Daniel Replogle

Abstract: Let $K_{m}=\Bbb{Q}(\zeta_{m})$ 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 $\Cal{O}_{m}=\Bbb{Z}[\zeta_{m}].$
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 $m$, $3 \leq m \leq 100,$ 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.

Timothy Kohl and Daniel Replogle <tkohl@bu.edu,dreplogle@cse.edu>