Cyclotomic Swan subgroups and primitive roots

Abstract

Let K m = Q ( ζ m ) where ζ m is a primitive mth root of unity. Let p > 2 be prime and let C p denote the group of order p . The ring of algebraic integers of K m is O m = Z [ ζ m ] . Let Λ m , p denote the order O m [ C p ] in the algebra K m [ C p ] . Consider the kernel group D ( Λ m , p ) and the Swan subgroup T ( Λ m , p ) . If ( p , m ) = 1 these two subgroups of the class group coincide. Restricting to when there is a rational prime p that is prime in O m requires m = 4 or q n where q > 2 is prime. For each such m, 3 ⩽ m ⩽ 100 , we give such a prime, and show that one may compute T ( Λ m , p ) as a quotient of the group of units of a finite field. When h mp + = 1 we give exact values for | T ( Λ m , p ) | , and for other cases we provide an upper bound. We explore the Galois module theoretic implications of these results.