Computation of several cyclotomic Swan subgroups

Let Cl(OK [G]) denote the locally free class group, that is the group of stable isomorphism classes of locally free OK [G]-modules, where OK 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(OK [G]), of Cl(OK [G]) when K = Q(p), p a primitive p-th root of unity, G = C2, where p is an odd (rational) prime so that h+ p = 1 and 2 is inert in K/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(Z[p]C2). These calculations give an alternative proof that the fields Q(p) for p=11, 13, 19, 29, 37, 53, 59, and 61 are not HilbertSpeiser.

• Type: Journal
• Year: 2002
• Where: MOC
• Authors: Timothy Kohl, Daniel R. Replogle
• Keywords: Free Class Group, MOC 2002, Primitive P-th Root, Stable Isomorphism Classes