Computation of several cyclotomic Swan subgroups

Timothy Kohl, Daniel R. Replogle
2000 Mathematics of Computation  
Let Cl(O K [G]) denote the locally free class group, that is the group of stable isomorphism classes of locally free O K [G]-modules, where 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, , ζp 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/Q. We show that, under these hypotheses, this calculation reduces to computing a quotient ring of a polynomial
more » ... g of a polynomial ring; we do the computations obtaining for several primes p a nontrivial divisor of Cl(Z[ζp]C 2 ). These calculations give an alternative proof that the fields Q(ζp) for p=11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.
doi:10.1090/s0025-5718-00-01302-8 fatcat:ufpumyj7yvdytn5s2ebvex7ywe