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 paper, it was shown that P(x) ≠ x is a cyclotomic polynomial with integer coefficients such that for infinitely many n, P(e 2πi n ) is a unit in the ring of algebraic integers.
6 citations
TL;DR: In this article, the primitive prime divisors of the terms of a real quadratic field and a unit element of its ring of integers were studied, and the methods used allow us to find the terms in the sequence that do not have a primitive prim divisor.
6 citations
Posted Content•
TL;DR: In this paper, it was shown that for any algebraic number field K of degree at least 3, there are only finitely many three times monogenic orders, and two special types of two-times-monogenic orders were defined.
Abstract: Let O be an order in an algebraic number field K, i.e., a ring with quotient field K which is contained in the ring of integers of K. The order O is called monogenic, if it is of the shape Z[w], i.e., generated over the rational integers by one element. By a result of Gy\H{o}ry (1976), the set of w with Z[w]=O is a union of finitely many equivalence classes, where two elements v,w of O are called equivalent if v+w or v-w is a rational integer. An order O is called k times monogenic if there are at least k different equivalence classes of w with Z[w]=O, and precisely k times monogenic if there are precisely k such equivalence classes. It is known that every quadratic order is precisely one time monogenic, while in number fields of degree larger than 2, there may be non-monogenic orders. In this paper we study orders which are more than one time monogenic. Our first main result is, that in any number field K of degree at least 3 there are only finitely many three times monogenic orders. Next, we define two special types of two times monogenic orders, and show that there are number fields K which have infinitely many orders of these types. Then under certain conditions imposed on the Galois group of the normal closure of K, we prove that K has only finitely many two times monogenic orders which are not of these types. We give some immediate applications to canonical number systems. Further, we prove extensions of our results for domains which are monogenic over a given domain A of characteristic 0 which is finitely generated over Z.
6 citations
TL;DR: In this article, it was shown that the set Cl ∞ ( O _ V ) of O { ∞ } -isomorphism classes in the genus of f of rank n > 2 is bijective as a pointed set to the abelian groups H et 2 ( O ∞, μ _ 2 ) ≅ Pic(C af ) / 2, i.e. it is an invariant of C af.
6 citations
TL;DR: In this paper, the authors studied integer solutions of an implicit linear inhomogeneous first order difference equation bxn+1 = axn + fn, and showed that for a = 1, a typical (in the natural topological sense) equation has no integer solutions.
Abstract: We study solutions in integers of an implicit linear inhomogeneous first order difference equation bxn+1 = axn + fn. Based on the p-adic topology on the ring of integers, we obtain a criterion for the existence of solutions and show that for a = 1 a typical (in the natural topological sense) equation has no integer solutions.
6 citations