Shin Nakano

Let l be the prime 3, 5 or 7 and let m be a nonzero integer. We give a method for constructing an infinite family of pairs of quadratic fields Q(root D) and Q(root mD) with both class numbers divisible by l. Such quadratic fields are parametrized by rational points on a specified elliptic curve.

We give a family of quintic cyclic fields with even class number parametrized by rational points on an elliptic curve associated with Emma Lehmer's quintic polynomial Further. we use the arithmetic of elliptic curves and the Chebotarev density theorem to show that there are infinitely many such fields. (C) 2009 Elsevier Inc All rights reserved.

Let p be an odd prime and r a divisor of p - 1. We present a characterization of metacyclic extensions of degree pr containing a given cyclic extension of degree r over a field of characteristic other than p. Furthermore, we give a method of constructing polynomials with Galois groups which are Frobenius groups of degree p. © 2002 by the University of Notre Dame. All rights reserved.

Let K/k be a Galois extension of a number field of degree n and p a prime number which does not divide n. The study of the p-rank of the ideal class group of K by using those of intermediate fields of K/k has been made by Iwasawa, Masley et al., attaining the results obtained under respective constraining assumptions. In the present paper we shall show that we can remove these assumptions, and give more general results under a unified viewpoint. Finally, we shall add a remark on the class numbers of cyclic extensions of prime degree of Q.

Using Cohen's construction of defining polynomials for a cyclic group of odd prime order, we define a polynomial with some parameters which generates cyclic extensions of a given odd prime degree, and prove it to be generic in the sense as defined below.

We investigate the relationship between the 2-class groups of cubic fields and the Mordell-Weil groups of elliptic curves defined over Q. There is an exact sequence which connects those two groups together for a certain kind of cyclic cubic fields. Using the twists of elliptic curves, we give a more precise result for the simplest cubic fields.