A note on the existence of certain infinite families of imaginary quadratic fields by Iwao Kimura (Toyama)1. Introduction. In this paper, we give a lower bound of the number of certain families of imaginary quadratic fields whose absolute value of discriminants are less than a given number (the main result is the corollary below). We briefly review the investigations concerned with this kind of problems.Let Z be the ring of rational integers, and Q the field of rational numbers. For any rational prime l, we denote by Z l the ring of l-adic integers. If k is an algebraic number field of finite degree over Q, we denote by h(k) the class number of k and by D(k) the discriminant of k. We let S denote the cardinality of any set S. Let (·/·) be the Legendre-Kronecker symbol.Denote by Q − the set of all imaginary quadratic fields and by Q + the set of all real quadratic fields. For any real number X > 0, let Q − (X) = {k ∈ Q − ; −X < D(k)} and Q + (X) = {k ∈ Q + ; D(k) < X}. Note that Q − (X) and Q + (X) are finite sets.Hartung [13] proved that {k ∈ Q − ; 3 h(k)} is an infinite set, and remarked that his method (an application of Kronecker's class number relation) can be applied to the same statement in which 3 is replaced by any odd prime l. The case of l = 3 is also implied in Davenport-Heilbronn's investigation [10,11]
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
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