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 published on a yearly basis
Papers
More filters
TL;DR: In this article, the authors considered the question of power bases in the cyclotomic number field Q(ζ), ζp = 1, p an odd prime, and showed that up to conjugacy, there can be no further integral generators for this order.
14 citations
01 Jan 2000
TL;DR: In this article, the authors construct a linear embedding of an associated module for a pair of ideals in an extension of a Dedekind ring into a tensor square of its fraction.
Abstract: Using the standard duality we construct a linear embedding of an associated module for a pair of ideals in an extension of a Dedekind ring into a tensor square of its fraction eld. Using this map we investi- gate properties of the coecien t-wise multiplication on associated orders and modules of ideals. This technique allows to study the question of de- termining when the ring of integers is free over its associated order. We answer this question for an Abelian totally wildly ramied p-extension of complete discrete valuation elds whose dieren t is generated by an element of the base eld. We also determine when the ring of integers is free over a Hopf order as a Galois module.
14 citations
TL;DR: In this paper, it was shown that the tropical variety of an ideal ideal over a field K with non-trivial discrete valuation can always be traced back to the tropical varieties of a Grobner polyhedra covering them.
Abstract: We show how the tropical variety of an ideal
$$I\unlhd K[x_1,\ldots ,x_n]$$
over a field K with non-trivial discrete valuation can always be traced back to the tropical variety of an ideal
$$\pi ^{-1}I\unlhd R\llbracket t\rrbracket [x_1,\ldots ,x_n]$$
over some dense subring R in its ring of integers. We show that this connection is compatible with the Grobner polyhedra covering them. Combined with previous works, we thus obtain a framework for computing tropical varieties over general fields with valuations, which relies on the existing theory of standard bases if
$$\pi ^{-1}I$$
is generated by elements in
$$R[t,x_1,\ldots ,x_n]$$
.
14 citations
TL;DR: In this article, the authors extend this approach to multiple coincidences, which apply to triple or multiple junctions, and give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions.
Abstract: Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions.
14 citations
TL;DR: In this paper, the authors considered the question of whether a line bundle on the arithmetic variety X is trivial when it is isomorphic to the structure sheaf OX and showed that the answer is always positive when X = P n.
Abstract: In this note, we will use the term “arithmetic variety” for a normal scheme X for which the structure morphism f : X → Spec(Z) is proper and flat. Let V be a proper, normal (not necessarily geometrically connected) variety over Q. Let us choose a normal model for V over Z, that is an arithmetic variety X whose generic fiber is identified with V . Suppose that F is a number field and consider the F -rational points of V . These correspond bijectively to R-valued points of X, with R the ring of integers of F .I fP is an F -rational point of V , we will also denote by P : Spec(R) → X the corresponding R-valued point of X. Suppose that L is a line bundle on the arithmetic variety X. We say that L is trivial, when it is isomorphic to the structure sheaf OX. We will denote by P ∗ L the pull-back of L to Spec(R) via the morphism P ; then P ∗ L is a line bundle on Spec(R). It gives an element (P ∗ L) in the class group Pic(R )o fR. In what follows, we will identify Pic(R) with the ideal class group Cl(F ). This paper is motivated by the following question of the second named author: Question. Suppose that the line bundle L on X is not trivial. Is there a number field F and an F -rational point P of V such that the ideal class (P ∗ L) is not trivial? As a variant of this question, we could also ask: Is there a scheme Z which is finite and flat over Spec(Z) and a morphism P : Z → X such that (P ∗ L )i s not trivial in Pic(Z)? L. Szpiro has informed us that he independently raised this question earlier. If the answer to the question is always positive, then line bundles on arithmetic varieties are characterized by their restrictions to integral points. Here are some interesting facts about this question: 1. The answer is positive when X = P n . Indeed, restricting along any linear morphism P 1 → P n gives an isomorphism on Picard groups. Therefore, it is enough to show the statement for X = P 1 . Take L = OX(n), n � 0. There is a number field F with an ideal A whose ideal class is not n-torsion. We can always write A = aOF + bOF with a, b in OF . Consider the F -rational point (a; b )o fP 1 . This gives a morphism P :S pec(OF ) → P 1F → P 1 .
14 citations