scholarly journals On D(w)-quadruples in the rings of integers of certain pure number fields

2014 ◽  
Vol 49 (1) ◽  
pp. 37-46
Author(s):  
Ljerka Jukić Matić
Keyword(s):  
Number Fields ◽  
Rings Of Integers ◽  
Pure Number

Canonical number systems, counting automata and fractals

2002 ◽  
Vol 133 (1) ◽  
pp. 163-182 ◽  
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.

ON THE CLASSIFICATION OF EVEN UNIMODULAR LATTICES WITH A COMPLEX STRUCTURE

2012 ◽  
Vol 08 (04) ◽  
pp. 983-992 ◽  
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.

scholarly journals The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields

2016 ◽  
Vol 152 (6) ◽  
pp. 1111-1120 ◽  
Author(s):  
Manjul Bhargava ◽  
Piper Harron
Keyword(s):  
Number Fields ◽  
Rings Of Integers

For$n=3$,$4$, and 5, we prove that, when$S_{n}$-number fields of degree$n$are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices.

scholarly journals ON INTEGRAL BASIS OF PURE NUMBER FIELDS

Mathematika ◽  
2020 ◽  
Vol 67 (1) ◽  
pp. 187-195
Author(s):  
Anuj Jakhar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Number Fields ◽  
Integral Basis ◽  
Pure Number

scholarly journals Two-primary algebraic $K$-theory of rings of integers in number fields

1999 ◽  
Vol 13 (1) ◽  
pp. 1-54 ◽  
Author(s):  
J. Rognes ◽  
C. Weibel ◽  
appendix by M. Kolster
Keyword(s):  
Number Fields ◽  
Rings Of Integers ◽  
K Theory
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