Divisibility of Orders of K2 Groups Associated to Quadratic Field

Abstract

Abstract. We discuss some divisibility results of orders of /f-groups and cohomology groups associated to quadratic fields. IntroductionIn our previous paper We raised, in above mentioned paper, the following question (see Question 2.4 for more precise statement): QUESTION 1.1. For any odd prime p, is there real quadratic field Q(\/D) such that p divides the numerator of CQ(^£>)(~1) Further, are there infinitely many such real quadratic fields?In §2, we answer this question affirmatively for primes p = 3,5. A similar problem for imaginary quadratic fields is considered in §3.We use the following notations throughout this paper. For any set S, jj5 is the cardinality of S. If K/k is an extension of fields, [K : fc] is the degree of K/k. If further K/k is a Galois extension, Gal(K/k) is its Galois group. As usual, Z is the ring of rational integers, Z>o is the set of non-negative rational integers.2000 Mathematics Subject Classification: 11R42, 11R70