Ratios of rings of integers as Galios modules

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.

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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

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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.

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Galois module structure of rings of integers

Journées Arithmétiques 1980 ◽

10.1017/cbo9780511662027.012 ◽

1982 ◽

pp. 218-225

Author(s):

M.J. Taylor

Keyword(s):

Module Structure ◽

Galois Module ◽

Galois Module Structure ◽

Rings Of Integers

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Factor Rings of Integers

American Mathematical Monthly ◽

10.1080/00029890.1989.11972234 ◽

1989 ◽

Vol 96 (6) ◽

pp. 521-522

Author(s):

Steve Johnson

Keyword(s):

Rings Of Integers

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ON THE CLASSIFICATION OF EVEN UNIMODULAR LATTICES WITH A COMPLEX STRUCTURE

International Journal of Number Theory ◽

10.1142/s1793042112500583 ◽

2012 ◽

Vol 08 (04) ◽

pp. 983-992 ◽

Cited By ~ 3

Author(s):

MICHAEL HENTSCHEL ◽

ALOYS KRIEG ◽

GABRIELE NEBE

Keyword(s):

Modular Group ◽

Complex Structure ◽

Theta Series ◽

Number Fields ◽

Quadratic Number ◽

Imaginary Quadratic Fields ◽

Rings Of Integers ◽

Hermitian Modular Form ◽

Imaginary Quadratic Number

This paper classifies the even unimodular lattices that have a structure as a Hermitian [Formula: see text]-lattice of rank r ≤ 12 for rings of integers in imaginary quadratic number fields K of class number 1. The Hermitian theta series of such a lattice is a Hermitian modular form of weight r for the full modular group, therefore we call them theta lattices. For arbitrary imaginary quadratic fields we derive a mass formula for the principal genus of theta lattices which is applied to show completeness of the classifications.

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