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Cyclotomic Swan subgroups and primitive roots | | Timothy Kohl
; Daniel Replogle
; | | Date: |
13 Nov 2002 | | Subject: | Number Theory | math.NT | | 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. | | Source: | arXiv, math.NT/0211468 | | Services: | Forum | Review | PDF | Favorites | |
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