Computation of several cyclotomic Swan subgroups

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 $K$ and $G$ 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 $p$-th root of unity, $G=C_{2}$, where $p$ 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 $p$ 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 $p$=11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.