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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.
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Dissertation•
01 Jan 1994
4 citations
TL;DR: In this article, it was shown that if a finite subgroup G of GLn(OK) is invariant under the action of Gal(K/Q ) then it is contained inGLn(Kab), where Kab is the ring of integers in a finite Galois extension of Q and Kabis the maximal abelian subextension of K.
4 citations
Posted Content•
TL;DR: In this paper, the authors consider the problem of constructing subsets of algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions with high upper density.
Abstract: A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.
4 citations
TL;DR: In this paper, it was shown that for every prime p, N1(Cp Cp) = 2p, and also proved that N 1(Cmp Cmp) = 1mp if n 1 cm Cm = 2m and m is large enough.
Abstract: Let K be an algebraic number field with non-trivial class group G and OK be its ring of integers. For k 2 N and some real x 1, let Fk(x) denote the number of non-zero principal ideals aOK with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that Fk(x) behaves, for x ! 1, asymptotically like x(logx) 1/|G| 1 (loglogx) Nk(G) . In this article, it is proved that for every prime p, N1(Cp Cp) = 2p, and it is also proved that N1(Cmp Cmp) = 2mp if N1(Cm Cm) = 2m and m is large enough. In particular, it is shown that for each positive integer n there is a positive integer m such that N1(Cmn Cmn) = 2mn. Our results confirm a conjecture given by W. Narkiewicz thirty years ago partly and improve the known results substantially.
4 citations
TL;DR: In this paper, an extensive study of the spaces of automorphic forms for of weight and level, for an ideal in the ring of integers of the quartic CM field of twelfth roots of unity, was performed through the computation of the Hecke module and the corresponding Hcke action.
Abstract: In this paper we perform an extensive study of the spaces of automorphic forms for of weight and level , for an ideal in the ring of integers of the quartic CM field of twelfth roots of unity. This study is conducted through the computation of the Hecke module , and the corresponding Hecke action. Combining this Hecke data with the Faltings–Serre method for proving equivalence of Galois representations, we are able to provide the first known examples of modular elliptic curves over this field. Supplementary materials are available with this article.
4 citations