Computation of several cyclotomic Swan subgroups期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Computation of several cyclotomic Swan subgroups

Authors:	Timothy Kohl Daniel R Replogle

Institution:	Office of Information Technology, Boston University, Boston, Massachusetts ; Department of Mathematics and Computer Science, College of Saint Elizabeth, Morristown, New Jersey

Abstract:	Let $Cl(\mathcal{O}_{K}G])$ denote the locally free class group, that is the group of stable isomorphism classes of locally free $\mathcal{O}_{K}G]$ -modules, where $\mathcal{O}_{K}$ is the ring of algebraic integers in the number field and is a finite group. We show how to compute the Swan subgroup, $T(\mathcal{O}_{K}G])$ , of $Cl(\mathcal{O}_{K}G])$ when $K=\mathbb{Q} (\zeta_{p})$ , $\zeta_{p}$ a primitive -th root of unity, $G=C_{2}$ , where is an odd (rational) prime so that $h_{p}^{+}=1$ and 2 is inert in $K/\mathbb{Q} .$ We show that, under these hypotheses, this calculation reduces to computing a quotient ring of a polynomial ring; we do the computations obtaining for several primes a nontrivial divisor of $Cl(\mathbb{Z} \zeta_{p}]C_{2}).$ These calculations give an alternative proof that the fields $\mathbb{Q} (\zeta_{p})$ for =11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.

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