Fuminori Kawamoto

We give a construction of positive integers with even period of minimal type.

We give a construction of positive integers with even period of minimal type.

The main purpose of this article is to present a numerical data which shows relations between real quadratic fields of class number 1 and a mysterious behavior of the period of simple continued fraction expansion of certain quadratic irrationals. For that purpose, we define a class number, a fundamental unit, a discriminant and a Yokoi invariant for a non-square positive integer, and then see that a generalization of theorems of Siegel and of Yokoi holds. These and a theorem of Friesen and Halter-Koch imply several interesting conjectures for solving Gauss' class number problem for real quadratic fields.

In a previous paper [4], we introduced the notion of real quadratic fields with period I of minimal type in terms of continued fractions. As a consequence, we have to examine a construction of real quadratic fields with period >= 5 of minimal type in order to find many real quadratic fields of class number 1. When l >= 4, it appears that there exist infinitely many real quadratic fields with period l of minimal type. Indeed, we provided an infinitude of real quadratic fields with period 4 of minimal type in [4]. In this paper, we construct an infinite family of real quadratic fields with large even period of minimal type whose class number is greater than any given positive integer, and whose Yokoi invariant is greater than any given positive integer.

In a previous paper [4], we introduced the notion of real quadratic fields with period I of minimal type in terms of continued fractions. As a consequence, we have to examine a construction of real quadratic fields with period >= 5 of minimal type in order to find many real quadratic fields of class number 1. When l >= 4, it appears that there exist infinitely many real quadratic fields with period l of minimal type. Indeed, we provided an infinitude of real quadratic fields with period 4 of minimal type in [4]. In this paper, we construct an infinite family of real quadratic fields with large even period of minimal type whose class number is greater than any given positive integer, and whose Yokoi invariant is greater than any given positive integer.

The main purpose of this article is to introduce the notion of real quadratic fields of minimal type in terms of continued fractions with period e. We show that fundamental units of real quadratic fields that are not of minimal type are relatively small. So, we see by a theorem of Siegel that such fields have relatively large class numbers. Also, we show that there exist exactly 51 real quadratic fields of class number 1 that are not of minimal type, with one more possible exception. All such fields are listed in the table of Section 8.2. Therefore we study real quadratic fields with period 9 of minimal type in order to find real quadratic fields of class number 1, and first examine the case where l <= 4. In particular we obtain a result on Yokoi invariants m(d) and class numbers h(d) of real quadratic fields Q(root d) with period 4 of minimal type.

We prove that a number field K satisfies the following property (B) if and only if the ray class group of K defined modulo 4 is trivial. (B): For any tame abelian extensions N-1 and N-2 over K of exponent 2, the composite N1N2/K has a relative normal integral basis (NIB) if both N-1/K and N-2/K have a NIB.

Let F be a number field. We construct three tamely ramified quadratic extensions K-i/F (1less than or equal toiless than or equal to3) which are ramified at most at some given set of finite primes, such that K-3 subset of K1K2, both K-1/F and K-2/F have normal integral bases, but K-3/F has no normal integral basis. Since Hilbert-Speiser's theorem yields that every finite and tamely ramified abelian extension over the field of rational numbers has a normal integral basis, it seems that this example is interesting (cf. [5] J. Number Theory 79 (1999) 164; Theorem 2). As we shall explain below, the previous papers (Acta Arith. 106 (2) (2003) 171-181; Abh. Math. Sem. Univ. Hamburg 72 (2002) 217-233) motivated the construction. We prove that if the class number of F is bigger than 1, or the strict ray class group Cl-4I0, of F modulo 4 has an element of order greater than or equal to3, then there exist infinitely many triplets (K-1, K-2, K-3) of such fields. (C) 2003 Elsevier Science (USA). All rights reserved.

Let F be a totally real number field and l a product of some real primes of F. J. B RINKHUIS gave a necessary condition for a finite abelian extension of F which is unramified outside l to have a normal integral basis. We consider the converse of his result and give a necessary and sufficient condition. Furthermore, we concretely express it when F is a real quadratic field or a cyclic cubic field.

Let F be a quadratic field and p a prime ideal in F. Then we ask whether the ray class field of F mod p has a normal integral basis over F. We see many differences between our case and the case where the base field F is the field of rational numbers. (C) 2001 Academic Press.