scispace - formally typeset
Search or ask a question
Topic

Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.


Papers
More filters
Journal ArticleDOI
TL;DR: In this paper , a bounded generation result concerning the minimum number of conjugates of suitable elementary matrices (or more precisely root elements) needed to write any element of the principal congruence subgroup of a ring of algebraic integers in any number field is given.
Abstract: For rings of algebraic integers [Formula: see text] in a number field [Formula: see text] called [Formula: see text]-pseudo-good, this paper describes a bounded generation result concerning the minimal number of conjugates of suitable elementary matrices (or more precisely root elements) in [Formula: see text] needed to write any element of the [Formula: see text]-principal congruence subgroup of [Formula: see text] as their product. Using this bounded generation result, we give explicit bounds for the diameter of word norms on [Formula: see text] given by conjugacy classes thereby continuing an investigation into such diameters by Kedra et al. Additionally, we present some examples of [Formula: see text]-pseudo-good rings and classify normally generating subsets of [Formula: see text] for [Formula: see text] the ring of algebraic integers in any number field.

3 citations

Journal ArticleDOI
TL;DR: In this paper, the weak Leopoldt conjecture and weak Jannsen conjecture for Tate twist realizations have been proved under some technical conditions, and the weak leopard conjecture has been shown to hold for almost all Tate twists.

3 citations

Journal ArticleDOI
TL;DR: In this article, a Galois extension of the number field K is considered, and a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in, the ring of integers of K, where β is the unique real satisyfing.
Abstract: Let L be a Galois extension of the number field K. Set n = nK = deg K/ℚ, nL = deg L/ℚ and nL/K = deg L/K. Let I = IL/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ1, λ2, …, λn − 1) be a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in , the ring of integers of K. Fixing an extension of each λi to a character on I, then λi,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define . Then the small region of K referred to above isfor 0

3 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the ring of integers of real quadratic fields which are sums of integral squares are in fact sums of distinct squares, provided their norm is large enough.

3 citations

Journal ArticleDOI
TL;DR: In this article, the polynomial Szemeredi theorem was generalized to intersective polynomials over the ring of integers of an algebraic number field, by which they mean polynoms having a common root modulo every ideal.
Abstract: We generalize the polynomial Szemeredi theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new polynomial configurations in positive-density subsets of and strengthens and extends recent results of Bergelson, Leibman and Lesigne [Intersective polynomials and the polynomial Szemeredi theorem. Adv. Math. 219(1) (2008), 369–388] on polynomials over the integers.

3 citations


Network Information
Related Topics (5)
Algebraic geometry
8.7K papers, 205K citations
89% related
Conjecture
24.3K papers, 366K citations
86% related
Elliptic curve
13.9K papers, 255.3K citations
86% related
Automorphism
15.5K papers, 190.6K citations
86% related
Polynomial
52.6K papers, 853.1K citations
85% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202250
2021117
2020121
2019111
201896