Stably Free and Not Free Rings of Integers

AbstractLet m = pe be a power of a prime number p. We say that a number field F satisfies the property when for any a ∈ F×, the cyclic extension F(ζm, a1/m)/F(ζm) has a normal p-integral basis. We prove that F satisfies if and only if the natural homomorphism is trivial. Here K = F(ζm), and denotes the ideal class group of F with respect to the p-integer ring of F.

Download Full-text

Representations of Cyclic Groups in Rings of Integers, I

Annals of Mathematics ◽

10.2307/1970266 ◽

1962 ◽

Vol 76 (1) ◽

pp. 73 ◽

Cited By ~ 42

Author(s):

A. Heller ◽

I. Reiner

Keyword(s):

Cyclic Groups ◽

Rings Of Integers

Download Full-text

Canonical number systems, counting automata and fractals

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004102005856 ◽

2002 ◽

Vol 133 (1) ◽

pp. 163-182 ◽

Cited By ~ 18

Author(s):

KLAUS SCHEICHER ◽

JÖRG M. THUSWALDNER

Keyword(s):

Direct Methods ◽

Number Fields ◽

Dimensional Vector ◽

Dimensional Vector Space ◽

Box Counting ◽

Number Systems ◽

Rings Of Integers ◽

Box Counting Dimension ◽

Self Similar ◽

Affine Set

In this paper we study properties of the fundamental domain [Fscr ]β of number systems, which are defined in rings of integers of number fields. First we construct addition automata for these number systems. Since [Fscr ]β defines a tiling of the n-dimensional vector space, we ask, which tiles of this tiling ‘touch’ [Fscr ]β. It turns out that the set of these tiles can be described with help of an automaton, which can be constructed via an easy algorithm which starts with the above-mentioned addition automaton. The addition automaton is also useful in order to determine the box counting dimension of the boundary of [Fscr ]β. Since this boundary is a so-called graph-directed self-affine set, it is not possible to apply the general theory for the calculation of the box counting dimension of self similar sets. Thus we have to use direct methods.

Download Full-text