The Tame Kernel of Multiquadratic Number Fields

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.