Reflection Theorems and the Tame Kernel of a Number Field

Abstract

Let L be a number field containing the pth primitive root of unity ζ p . We investigate the p-rank of the ideal class groups of some subfields of L by using reflection theorems and establish relations between the p-rank of the ideal class groups and that of groups of units of some subfields of L. Let F be a number field and 𝒪 F the ring of integers in F. We also study the p-rank of tame kernels of F and establish relations between the p-rank of K 2𝒪 F and that of some direct summands of the ideal class group of F(ζ p ).