Swan Modules and Realisable Classes for Kummer Extensions of Prime Degree

✍ Scribed by Daniel R Replogle

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 118 KB
• Volume: 212
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7652

✦ Synopsis

Let l ) 2 be a prime number. Let O O denote a ring of algebraic integers of a F number field F and let G be the cyclic group of order l. Consider the ring of w x integers O O as a locally free O O G -module where LrK is a tamely ramified L K Galois extension of number fields with Galois group isomorphic to G. The classes of rings of integers obtained in such a way for a fixed number field K containing the lth roots of unity form the subgroup of realisable classes R in the locally free Ž w x. class group Cl O O G . On the other hand, one may also look at the Swan K Ž w x. w x Ž . subgroup T of Cl O O G , formed by the classes of locally free O O G -ideals s, ⌺ K K where s in O O is relatively prime to l and ⌺ denotes the sum of the elements of G. K Ž ly1.r2 Ž w x. We show that T ; R l D, where D is the kernel group of Cl O O G . We K also determine necessary and sufficient conditions for when a realisable class is a Ž . Swan class. Last, we show that R l D is nontrivial for K s Q when l ) 3 by l providing a nontrivial lower bound for the size of the Swan subgroup T.