Fuminori Kawamoto

On normal integral bases of local fields

(ii) Let k be a number field and K/k be a finite tamely ramified Galois extension with Galois group G. Then oK is a locally free o,Gmodule, where oK is the ring of integers of K and o,G is the group ring of G over ok. The fact (ii) is important on the normal integral basis problem of number fields (cf. Frohlich [2]). (ii) is the global version of (i). In the present paper, we shall give a simple proof of (i) without using representation theory, which will give a more explict form of n.i.b. (Note that if K/k is a finite Galois extension of number field or local field with an n.i.b., then K/k is tamely ramified.)

Continued fractions with even period and an infinite family of real quadratic fields of minimal type

In a previous paper [4], we introduced the notion of real quadratic fields with period l 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.

Normal integral bases and strict ray class groups modulo 4

Abstract Let F be a number field. We construct three tamely ramified quadratic extensions K i /F (1⩽i⩽3) which are ramified at most at some given set of finite primes, such that K3⊂K1K2, both K1/F and K2/F have normal integral bases, but K3/F has no normal integral basis. Since Hilbert–Speisers 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 4 l 0 of F modulo 4 has an element of order ⩾3, then there exist infinitely many triplets (K1,K2,K3) of such fields.

Normal integral bases of ∞ -ramified abelian extensions of totally real number fields

LetF be a totally real number field and [ a product of some real primes ofF. J. Brinkhuis gave a necessary condition for a finite abelian extension ofF which is unramified outside [ 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 whenF is a real quadratic field or a cyclic cubic field.