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.

The integer points on three related elliptic curves

Abstract: The integer points on the three elliptic curves y2 = 4cx3 + 13, c = 1, 3, 9, are found, with an application to coding theory. It is also shown that there are precisely three nonisomorphic cubic extensions of the rationals with discriminant -35 13. 1. In [I] the Diophantine equation (1) y2 = 4 . 3k + 13 is shown to arise from coding theory, and its integer solutions are found. By considering congruence classes of k modulo 3, this equation gives rise to the three elliptic curves (2) y2=4x3 + 13, (3) y2 = 12x3 + 13, (4) y2= 36x3 + 13. We find here all integral solutions of (2), (3), (4), giving as a corollary all solutions to Eq. (1). 2. Since Q( 13) has class number 1, Eq. (2) immediately reduces to an equation y+ 13 1+ 13 2 = E a+ b 2 J where a, b E Z, e (3 + 13)/2 is a fundamental unit of Q( 13), and where without loss of generality K0, + 1. Since a3 E Z[ 13 1 for every integer a E Q(133), the case K 0 is impossible. Comparing coefficients of 13 in the two cases K 1 gives respectively (5) K = 1: 1 -a3 + 6a2b + 15ab2 + Ilb3, (6) K -1: 1 a3 3a2b + 6ab2 b3. Under the respective substitutions (A, B) = (a + 2b, b), (A, B) = (a b, -b) both (5) and (6) reduce to (7) 1 =A3 + 3AB2-3B3. We now work in Q(X), where 23 + 3X 3 0 O. It is straightforward to verify that the ring of integers in this field is Z[X], and a fundamental unit is = X. (The Received May 8, 1981; revised October 13, 1981. 1980 Mathematics Subject Classificatioti. Primary IOB 10; Secondary 1 2A30. ? 1982 American Mathematical Society 0025-5718/81 /0000-0446/$0 1.75

On dedekind's criterion and monogenicity over dedekind rings

Abstract: We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ring R, based on results on the resultant Res(P , Pi) of the minimal polynomial P of a primitive integral element and of its irreducible factors Pi modulo prime ideals of R. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996) and we give some applications in the case where R is a discrete valuation ring or the ring of integers of a number field, generalizing some well-known classical results.

The Iwasawa Invariants and the Higher K-Groups Associated to Real Quadratic Fields

Abstract: Using fast algorithms, we compute the Iwasawa invariants of Q( √ f, ζp) in the range 1

Embedding Orders Into Central Simple Algebras

Abstract: The question of embedding fields into central simple algebras $B$ over a number field $K$ was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields $L$ of such an algebra into orders in that algebra is more nuanced. The first such result along those lines is an elegant result of Chevalley \cite{Chevalley-book} which says that with $B = M_n(K)$ the ratio of the number of isomorphism classes of maximal orders in $B$ into which the ring of integers of $L$ can be embedded (to the total number of classes) is $[L \cap \widetilde K : K]^{-1}$ where $\widetilde K$ is the Hilbert class field of $K$. Chinburg and Friedman (\cite{Chinburg-Friedman}) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona \cite{Arenas-Carmona} considers embeddings of the ring of integers into maximal orders in a broad class of higher rank central simple algebras. In this paper, we consider central simple algebras of dimension $p^2$, $p$ an odd prime, and we show that arbitrary commutative orders in a degree $p$ extension of $K$, embed into none, all or exactly one out of $p$ isomorphism classes of maximal orders. Those commutative orders which are selective in this sense are explicitly characterized; class fields play a pivotal role. A crucial ingredient of Chinberg and Friedman's argument was the structure of the tree of maximal orders for $SL_2$ over a local field. In this work, we generalize Chinburg and Friedman's results replacing the tree by the Bruhat-Tits building for $SL_p$.