Swan Modules and Realisable Classes for Kummer Extensions of Prime Degree

Letl > 2 be a prime number. Let Fdenote a ring of algebraic integers of a number fieldFand letGbe the cyclic group of orderl. Consider the ring of integers Las a locally free K[G]-module whereL/Kis a tamely ramified Galois extension of number fields with Galois group isomorphic toG. The classes of rings of integers obtained in such a way for a fixed number fieldKcontaining thelth roots of unity form the subgroup of realisable classesRin the locally free class groupCl( K[G]). On the other hand, one may also look at the Swan subgroupTofCl( K[G]), formed by the classes of locally free K[G]-ideals (s, Σ) wheresin Kis relatively prime toland Σ denotes the sum of the elements ofG. We show thatT(l − 1)/2 R ∩ D, whereDis the kernel group ofCl( K[G]). We also determine necessary and sufficient conditions for when a realisable class is a Swan class. Last, we show thatR ∩ Dis nontrivial forK = Q(ζl) whenl > 3 by providing a nontrivial lower bound for the size of the Swan subgroupT.