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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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TL;DR: In this paper, the authors studied the non-existence problem of (nearly) perfect (almost) $m$-ary sequences via their connection to Butson-Hadamard (BH) matrices and (near) conference matrices.
Abstract: In this paper we study the non-existence problem of (nearly) perfect (almost) $m$-ary sequences via their connection to (near) Butson-Hadamard (BH) matrices and (near) conference matrices. Firstly, we apply a result on vanishing sums of roots of unity and a result of Brock on the unsolvability of certain equations over a cyclotomic number field to derive non-existence results for near BH matrices and near conference matrices. Secondly, we refine the idea of Brock in the case of cyclotomic number fields whose ring of integers is not a principal ideal domains and get many new non-existence results.

7 citations

Journal ArticleDOI
TL;DR: In this article, the authors study the Hurwitz product (convolution) in the space of formal Laurent series over an arbitrary field of zero characteristic and obtain the convolution equation which is satisfied by the Euler series.
Abstract: We study the Hurwitz product (convolution) in the space of formal Laurent series over an arbitrary field of zero characteristic. We obtain the convolution equation which is satisfied by the Euler series. We find the convolution representation for an arbitrary differential operator of infinite order in the space of formal Laurent series and describe translation invariant operators in this space. Using the p-adic topology in the ring of integers, we show that any differential operator of infinite order with integer coefficients is well defined as an operator from ℤ[[z]] to ℤp[[z]].

7 citations

Journal ArticleDOI
TL;DR: If p≡1 ( mod 4) is an odd prime and q =2 p −1 is a prime power, then there exists a family of Hadamard matrices of dihedral group type and this theorem is proved by giving the elements of Z [ D 2 p ] concretely.

7 citations

Journal ArticleDOI
01 Jan 1999
TL;DR: In this article, the authors considered the problem of finding the set of k-tuples (al,..., ak) in a ring of integers in a number field for which 2a 2b divides 3a -3b, and showed that there are exactly 14 solutions.
Abstract: For given elements a1,... , ak and 3 belonging to the ring of integers A of a number field we consider the set of all k-tuples (al,... , ak) in Nk for which Ek l caiai divides Ekcaiai for any z E A, and prove under some reasonable assumptions that the set of solutions is finite. The original motivation for this work comes from a problem raised by J. L. Selfridge (see Guy [1], problem B47) who asks for what pairs (a, b) does 2a-2b divide nanb for all integers n. A related (but more difficult) problem proposed by H. Ruderman asks to show that if 2a 2b divides 3a -3b, then 2a 2b divides n nb for all integers n. This was investigated by B. Velez in [6]. While Ruderman's problem is still open, Selfridge's problem was solved by Pomerance [2], who combined results of Schinzel [4] with Velez's work. It turns out that there are exactly 14 solutions. The problem was also solved by Sun Qi and Zhang Ming Zhi [5]. In this paper we show that the above finiteness result is a particular case of a more general phenomenon. Let IC be a number field, A = AKC its ring of integers and U = UKc its group of units. Let c1,... , ak and p be nonzero elements of A. We consider the set of all k-tuples (al,... , ak) in Nk for which

7 citations

Journal ArticleDOI
TL;DR: In this article, the ring of integers of a quartic number field has a power basis, and it is shown that when the quadratic subfield of the field is imaginary, the problem is completely solved.

7 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202250
2021117
2020121
2019111
201896