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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: Esnault et al. as mentioned in this paper proved that if the model X is regular, one has a congruence |x(k)| ≡ 1 modulo |k| for the number of k-rational points.
Abstract: If the l-adic cohomology of a projective smooth variety, defined over a local field K with finite residue field k, is supported in codimension > 1, then every model over the ring of integers of K has a k-rational point. For K a p-adic field, this is proved in (Esnault, 2007, Theorem 1.1). If the model X is regular, one has a congruence |x(k)| ≡ 1 modulo |k| for the number of k-rational points (Esnault, 2006, Theorem 1.1). The congruence is violated if one drops the regularity assumption.
7 citations
TL;DR: In this paper, it was shown that a matrix in a hereditary R-order in a central separable algebra over a global field K with ring of integers R, having determinant divisible by r∈R has a left factor having determinants r having reduced norm an associate of rϵR whenever r divides the reduced norm of λ.
7 citations
01 Mar 1993
TL;DR: For any two lattices L and K in the same genus there exist isometric primitive sublattices L', K' of codimension 1 as discussed by the authors, and this result not only extends Friedland's conjecture but also extends it to lattices in an arbitrary genus and defined over any algebraic number field.
Abstract: For any two lattices L and K in the same genus there exist isometric primitive sublattices L' , K' of codimension 1. This result not only proves Friedland's conjecture but also extends it to lattices in an arbitrary genus and defined over any algebraic number field. In [F] Friedland proved that for any two positive definite integral symmetric matrices A and B of rank n and determinant unity there are matrices A1 = (aij) and B1 = (b1j) which are Z-equivalent (as quadratic forms) to A and B respectively and such that aij = bij for 1 < i, j < n 2. He further conjectured that the same holds with 1 < i, j < n 1. His proof is matricial. We shall give here not only a lattice-theoretic proof of his theorem, but also a more general result not restricted to unimodular lattices rather for any genus and defined over any number field. Furthermore, we shall also prove the conjectured codimension-one result in this full general setting. In the course of the proof we shall first present a weaker version (Proposition) which is based on the theorem of "spinor linkage" [BH]. This version is weaker because the constructed pair of isometric sublattices in general has codimension two and will only have codimension one when the given lattices corresponding to the matrices A and B belong to the same spinor genus. However, this version has the virtue that the isometric sublattices are more effectively determinable since the effectiveness of finding the prime spot in the construction of the associated graph had been demonstrated in [BH]. The more general statement in the structure theorem below uses the stronger property of "class linkage" which is of independent interest. In what follows, let F be an algebraic number field, R its ring of integers, and V a regular quadratic space over F of dimension n. Let L and K be two R-lattices on V belonging to the same genus. L and K are said to be q-neighbors (for some discrete spot q on F) if Lp = Kp for p :# q and [Lq: Lq n Kq] = [Kq: Lq n Kq] = Norm(q). The "spinor genera linkage theorem" of [BH] states that we can always effectively find a discrete spot q away from 2 vol(L) and a lattice K' belonging to spn+K so that L and K' are Received by the editors September 3, 1991 and, in revised form, March 23, 1992. 1991 Mathematics Subject Classification. Primary 1 lE12; Secondary 15A63, 1 1D09, 11 E20. Research partially supported by N.S.F. ? 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page
6 citations
TL;DR: In this paper, the problem of deciding if a prime p divides k and if p I k is a prime in a monic irreducible polynomial in Z[x] was studied.
Abstract: Letf(x) be a monic irreducible polynomial in Z[x], and r a root of f(x) in C. Let K be the field Q(r) and X the ring of integers in K. Then for some k E Z, disc r = k2 disc M. In this paper we give constructive methods for (a) deciding if a prime p divides k, and (b) if p I k, finding a polynomial g(x) E Z[x] so that g(x) i 0 (mod p) but g(r)/p E M.
6 citations
TL;DR: In this article, it was shown that if R is a Hilbert ring and every maximal ideal in R has height >n, then every maximal ideals in R[XJ has height ≥n + 1.
6 citations