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 ).

Key Words:

• Ideal class groups
• Tame kernels

2010 Mathematics Subject Classification:

• Primary 11R70
• Secondary 11R16, 11Y40, 19-04, 19C99

ACKNOWLEDGMENTS

The authors would like to express their sincere gratitude to the referees for their careful reading of the manuscript and suggestions on the writing of the paper. This work was supported by NSFC (Nos. 11301071, 11326052, 11171141, 11271177, 11071110), Jiangsu Planned Projects for Postdoctoral Research Funds (Nos. 1202101c) and China Postdoctoral Science Foundation (Nos. 2013M531244).

Notes

Communicated by Q. Wu.