The Tame Kernel of Multiquadratic Number Fields

Abstract

Let F/K be a Galois extension of a number field of degree n, 𝒪 _F the ring of integers in F, and p a prime number which does not divide n. Let K ₂ denote the Milnor K-functor. In this article, we shall study the structure of the odd part of the tame kernel K ₂𝒪 _F of F by using the intermediate fields of F/K. In particular, for a multiquadratic field F, we shall get the p ⁱ -rank, (i > 0) of K ₂𝒪 _F . Finally, we shall determine the structure of the odd parts of K ₂𝒪 _F when where − 100 < d < 0, d ₁ = 2,3,5,7.

Key Words:

• Multiquadratic fields
• Tame kernels
• Wild kernels

2000 Mathematics Subject Classification:

• 11R70
• 11R21
• 19F99

ACKNOWLEDGMENT

I would like to thank the referee for his valuable comments. This article was supported by the NSFC 10801076 and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 08KJB110006).

Notes

Communicated by C. Pedrini.