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.

Integer Solutions of Matrix Linear Unilateral and Bilateral Equations over Quadratic Rings

Abstract: For matrix linear equations AX + BY = C and AX + YB = C over quadratic rings $$ \mathbb{Z}\left[\sqrt{k}\right] $$ , we establish necessary and sufficient conditions for the existence of integer solutions, i.e., solutions X and Y over the ring of integers $$ \mathbb{Z} $$ . We also present the criteria of uniqueness of the integer solutions of these equations and the method for their construction.

On the representation of integers as linear combinations of consecutive values of a polynomial

Abstract: Let K be a cyclotomic field with ring of integers O K and let f be a polynomial whose values on *Z belong to O K . If the ideal of O K generated by the values of f on Z is O K itself, then every algebraic integer N of K may be written in the following form: N = Σe k f(k) for some integer l, where the e k 's are roots of unity of K. Moreover, there are two effective constants A and B such that the least integer l (for a fixed N) is less than A N + B, where Formula math.

Left Zero Divisor Graphs of Totally Ordered Rings

Abstract: In this paper we consider prime graph of R (denoted by ) of an associative ring R (introduced by Satyanarayana, Syam Prasad and Nagaraju [6]). This short paper is divided into two Sections. Section-1 is devoted for preliminary definitions. In section-2, we constructed Left zero divisor graph of R (denoted by LZDG(R)) where R = the set of all matrices over the ring of integers modulo 2.

On the First Case of the Fermat Theorem for Cyclotomic Fields

Abstract: has no nonzero solutions in K. Let K be a finite extension of the rational number field Q (number field). One says that the first case of Fermat theorem (1CFT) holds for K and l if Eq. (1) has no integral solutions in K that are relatively prime to l. A remarkable achievement of the last few years is the proof of a problem that was standing for more than three hundred years, the FT for Q and all l > 3 (see [15]). Meanwhile, the analysis of solutions to Fermat equations in wider domains than Q, namely in its finite (or even infinite) extensions, is, apparently, still a more difficult and intriguing problem. The present paper deals with a generalization of the classical criteria of Kummer, Mirimanoff, and Vandiver for the validity of the CFT for Q and l. In brief, we prove the fulfillment of the 1CFT for L and l under the same assumptions on l that enter the corresponding classical criterion. Here L is the cyclotomic field generated over Q by the lth roots of unity. A number of authors simplified the sufficient conditions on l in the above-mentioned criteria. As a consequence, the corresponding criteria for fulfillment of the 1CFT for Q and l arose. For example, Wieferich’s criterion [14]: the 1CFT holds for Q and l if l2 does not divide 2−2. Certainly, these criteria for Q became of no importance because of the complete proof of the FT for Q by other methods. Our result implies that the 1CFT holds for L and l under the relevant conditions on l. In particular, the 1CFT holds for L and l if l2 does not divide 2 − 2. Let us state the obtained results in detail. Let ζ be a primitive lth root of unity. Let O be the ring of integers of the field L = Q(ζ), and let α = (ζ − 1)O be a unique prime ideal of O containing l. The residue field O/α can be identified with Z/lZ via the canonical isomorphism Z/lZ −→ O/α. Let (x, y, z) be a solution to Eq. (1) in O\α. Let G = G(x,y,z) be the subset of Z/lZ consisting of the classes (mod α) of the elements −x/y, −x/z, −y/x, −y/z, −z/x, −z/y. Since x+y+z ≡ x+y+z = 0 (mod α), the set G contains neither 0 nor 1. If t ∈ G, then G is exactly the set {t, 1− t, 1/t, 1/(1 − t), t/(t− 1), 1− 1/t}. Let rational numbers Bn be defined by the expansion

Class invariants for tame Galois algebras

Abstract: Let G be a finite group and K a number field with ring of integers O_K. In this thesis we study several questions related to the locally free class group Cl(O_K[G]). We mainly focus on the investigation of the set of classes in Cl(O_K[G]) which can be obtained from the ring of integers of tame G-Galois extensions of K. This set R(O_K[G]) is called the set of realizable classes. When G is abelian, R(O_K[G]) equals the Stickelberger subgroup St(O_K[G]); when G is not abelian, we just know that R(O_K[G]) is contained in St(O_K[G]), while the question if R(O_K[G]) is a group is still open. In this dissertation, after a general introduction to the subject and an exposition of the results mentioned above, we prove that St(O_K[G]) is trivial, if K is the field of rational numbers and G is a cyclic group of prime order or the dihedral group of order 2p, with p an odd prime number. Afterwards we study the functorial behavior of St(O_K[G]) under base field restriction. In the last part, restricting our attention to the abelian case, we give a result concerning the distribution of the Galois structures of the ring of integers of tame G-Galois extensions of K among realizable classes.