Cyclotomic Swan subgroups and primitive roots

Details Cyclotomic Swan subgroups and primitive roots Cyclotomic Swan subgroups and primitive roots

2002-11-13

arxiv.org/abs/math/0211468v1

Mathematics

Number Theory

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Cyclotomic Swan subgroups and primitive roots does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Cyclotomic Swan subgroups and primitive roots, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cyclotomic Swan subgroups and primitive roots will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-294140