Showing papers on "Ring of integers published in 1972"
TL;DR: Given an integer m, a product of distinct primes p i, a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF ( p i ).
Abstract: Given an integer m which is a product of distinct primes p i , a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF ( p i ). Specifically, if we are given a cyclic ( n , k i ) code over GF ( p i ) with minimum Hamming distance d i , for each i , then we construct a code of block length n over the integers modulo m with π i p k i i codewords, which is both linear and cyclic and has minimum Hamming distance min i d i .
124 citations
TL;DR: In this article, the relationship between values of zeta and L-functions and Euler characteristics of sheaves for the etale topology was studied, and it was shown how to identify a sheaf for the topology on SpecF.
Abstract: In this paper we begin a study of the relationship between values of zeta and L-functions on the one hand, and Euler characteristics of sheaves for the etale topology, on the other hand. We start with the following conjecture: Let F be a totally real number field, p a prime, and n a negative odd integer. Let A be the ring of integers of F, S the set of primes of F over p, X = SpecA, and X, = X S. Let j be the natural inclusion of SpecF in X, Let F be the algebraic closure of F, and GF the Galois group of F over F. Let W be the GF-module consisting of all p-power roots of unity in F, let T(W) be the Tate-module of W, and let W(m' be the GF-module W ? T(W)xm('". Then W(m' may be identified with a sheaf for the etale topology on SpecF, and we may take the direct image j* W(m' on X, If M is a finite group, let #(M) denote the number of elements of M.
58 citations
TL;DR: In this article, the dimension subgroup conjecture is shown to be true for the general ring of integers modulo the prime power p. The dimension subgroups are characterized group-theoretically; they are not generally the appropriate subgroups but are seen to be closely related.
23 citations
TL;DR: The class number of an arbitrary o-lattice on a quadratic F-space in terms of a computable set of invariants remains a major open problem, although some progress has been made in special cases.
Abstract: The Hasse-Minkowski Theorem reduces the isometry question for quadratic spaces over global fields to the corresponding problem over local fields. But the analogous result does not hold in the integral situation. Thus if F is an algebraic number field and o is the ring of integers of F, then two o-lattices on a quadratic F-space V may be in the same genus (that is, locally isometric at all primes p) while not in the same isometry class. The determination or estimation of the class number (the number of classes in the genus) of an arbitrary o-lattice on a quadratic F-space in terms of a computable set of invariants remains a major open problem, although some progress has been made in special cases. For example, it has been shown (see
15 citations
TL;DR: The papers [1] and [5] by Gandy and Mendelson show that R is not a nonstandard model of Z but the paper given here is a simpler proof based on Mendelson's paper.
Abstract: In [3] Kemeny made the following conjecture: Suppose * Z is a nonstandard model of the ring of integers Z . Let and let F be the subgroup of those cosets ā which contain an element of infinite height in * Z . Kemeny then asked if the ring R = { a: ā ∈ F } is also a nonstandard model of Z . If so then Goldbach's conjecture is false because Kemeny also shows in [3] that Goldbach's conjecture fails in R . The papers [1] and [5] by Gandy and Mendelson show that R is not a nonstandard model of Z but we give here a simpler proof based on Mendelson's paper. Suppose R is a nonstandard model of Z . Then each positive number in R is a sum of four squares. Choose a in R so that a is a positive element of R of infinite height in * Z . Then since a is infinite in * Z , a − 1 is positive. Thus , x i ∈ R for i = 1, …, 4. Now each x i must be of the form a i + n i , where a i has infinite height in * Z and n i , ∈ Z .
2 citations
TL;DR: In this paper, the number of spinor genera in a genus is examined and shown to be finite in all cases where the genus does not contain a quaternary anisotropic subspace.
1 citations
TL;DR: In this paper, it was shown that the integral quadratic form Q(X, Y) = aX2 + bXY + cY2, a, b, c ∈ k(x), and X and Y taking values in k[x] is in the same genus.