Algebraic Number Field Research Articles

Using the Interval-Symbol Method with Zero Rewriting to Factor Polynomials over Algebraic Number Fields

The ISZ method (Interval-Symbol method with Zero rewriting) based on stabilization theory was proposed to reduce the amount of exact computations as much as possible but obtain the exact results by aid of floating-point computations. In this paper, we applied the ISZ method to Trager's algorithm which factors univariate polynomials over algebraic number fields. By Maple experiments, we show the efficiency of the ISZ method over the purely exact approach which uses exact computations throughout the execution of the algorithm. Furthermore, we propose a new method called the ISZ* method, which is similar to the ISZ method but beforehand excludes insufficient precisions of floating-point approximation by checking the correctness of the obtained supports. We confirmed that the ISZ* method is more effective than the ISZ method when the initially set precision is not sufficiently high.

History and development of algebraic number theory

We briefly describe the origin and development of classical Algebraic Number Theory. We state some latest results regarding discriminants and integral bases of pure number fields as well as sextic number fields defined by irreducible trinomials with integer coefficients. A few open problems regarding class number of algebraic number fields are also mentioned.

Bounds for sets with few distances distinct modulo a prime ideal

Let 𝒪 K be the ring of integers of an algebraic number field K embedded into ℂ. Let X be a subset of the Euclidean space ℝ d , and D(X) be the set of the squared distances of two distinct points in X. In this paper, we prove that if D(X)⊂𝒪 K and there exist s values a 1 ,...,a s ∈𝒪 K distinct modulo a prime ideal 𝔭 of 𝒪 K such that each a i is not zero modulo 𝔭 and each element of D(X) is congruent to some a i , then |X|≤d+s s+d+s-1 s-1.

Polynomial continued fractions for exp(π)

ABSTRACT We present two (inequivalent) polynomial continued fraction representations of the number with all their elements in ; no such representation was seemingly known before. More generally, a similar result for is obtained for every such that . The proof uses a classical polynomial continued fraction representation of , for and , of which we present a new proof that enables us to obtain the exact rate of convergence of the convergents of the continued fraction for . We also deduce some consequences of arithmetic interest concerning the elements of certain polynomial continued fraction representations of the (transcendental) Gel'fond-Schneider numbers , where and , where is the field of algebraic numbers, embedded into .

Matrix-valued Schrödinger operators over finite adeles

Let [Formula: see text] be an algebraic number field. With [Formula: see text] we associate the ring of finite adeles [Formula: see text] In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group [Formula: see text] Specifically, we consider matrix-valued Hamiltonian operators [Formula: see text] where [Formula: see text] is the Vladimirov operator and [Formula: see text] is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by [Formula: see text] This formula is given in terms of the ordered time exponentials.

ON NONMONOGENIC ALGEBRAIC NUMBER FIELDS

Let q be a prime number and f(x)=xqs−axm−b be a monic irreducible polynomial of degree qs having integer coefficients. Let K=ℚ(𝜃) be an algebraic number field with 𝜃 a root of f(x). We give some explicit conditions involving only a,b,m,q,s for which K is not monogenic. As an application, we show that if p is a prime number of the form 32k+1, k∈ℤ and 𝜃 is a root of a monic polynomial f(x)=x2s−32cpx2r−p∈ℤ[x] with s>4,2∤c,s≠5+r, then ℚ(𝜃) is not monogenic.

Non-monogenity of some number fields generated by binomials or trinomials of prime-power degree

Let [Formula: see text] be a prime number and [Formula: see text] be an algebraic number field with [Formula: see text] a root of an irreducible polynomial [Formula: see text] having integer coefficients. In this paper, we provide some explicit conditions involving only [Formula: see text] for which [Formula: see text] is non-monogenic. As an application, in the special case of [Formula: see text] and [Formula: see text], we show that if [Formula: see text] and [Formula: see text] divides [Formula: see text], then [Formula: see text] is not monogenic. We illustrate our results through examples.

On non-monogenity of the number fields defined by certain quadrinomials

Let p be a rational prime. Let be an algebraic number field with θ a root of an irreducible quadrinomial with . In the present paper, we give some explicit conditions involving only and s for which K is non-monogenic. If q is a prime number of the form , then in the special case when p = 2 and with , s > 4, we show that K is non-monogenic. Finally, we illustrate our results through examples.

Special solutions for the Laplace and diffusion equations associated with the algebraic number field

This article is devoted to the even entire functions, which are the exact solutions for the Laplace and diffusion equations. These functions are considered in the algebraic number field. We guess that the functions have purely real zeros in the entire complex plane. These are proposed as new connections with algebraic number theory and mathematical physics.

Non-monogenity of certain octic number fields defined by trinomials

Let $K=\mathbb Q(\theta )$ be an algebraic number field with $\theta $ a root of an irreducible polynomial $f(x)=x^8+ax^m+b\in \mathbb Z[x]$ and $1\leq m \leq 7$. We study the monogenity of $K$. Precisely, we give some explicit conditions on $a,b$ for whi

On a diffusion on finite adeles and the Feynman-Kac integral

Let K be an algebraic number field. With K, we associate the ring of finite adeles AK. Following a recent result of Weisbart on diffusions on finite rational adeles AQ, we define the Vladimirov operator ΔAK on AK and define the Brownian motion on the group AK. We also consider the Schrödinger operator −HAK=−ΔAK+V with a potential operator V given by a non-negative continuous function v on AK. We prove a version of the Feynman–Kac formula for the Schrödinger semigroup generated by −HAK.

Companions of fields of rational and real algebraic numbers

Companions of the field of rational numbers and a real-closed algebraic expansion of the field of rational numbers are studied. The description of existentially closed companions of a real-closed algebraic expansion of a field of rational numbers refers to the field of study of classical algebraic structures. The general theory of companions and existentially closed companions, built on the basis of Fraisse's classes in the works of A.T. Nurtazin, is included in the classical field of existentially closed theories in model theory. The basic concept of a companion: two models of the same signature are called companions if for any finite submodel of one of them, there is an isomorphic finite submodel in the other. This approach, applied to specific classical structures and their theories, provides new tools for the study of these objects. The study of the companion class of rational and algebraic real number fields reveals companion fields containing transcendental and possibly algebraic elements with special properties of polynomials defining these elements.

Common index divisor of the number fields defined by

AbstractLet $K={\mathbf {Q}}(\theta )$ be an algebraic number field with $\theta$ a root of an irreducible polynomial $x^5+ax+b\in {\mathbf {Z}}[x]$. In this paper, for every rational prime $p$, we provide necessary and sufficient conditions on $a,\,~b$ so that $p$ is a common index divisor of $K$. In particular, we give sufficient conditions on $a,\,~b$ for which $K$ is non-monogenic. We illustrate our results through examples.

Algebraic number fields generated by an infinite family of monogenic trinomials

For an infinite family of monogenic trinomials \(P(X)=X^3\pm 3rbX-b\in {\mathbb {Z}}[X]\), arithmetical invariants of the cubic number field \(L={\mathbb {Q}}(\theta )\), generated by a zero \(\theta\) of \(P(X)\), and of its Galois closure \(N=L(\sqrt{d_L})\) are determined. The conductor \(f\) of the cyclic cubic relative extension \(N/K\), where \(K={\mathbb {Q}}(\sqrt{d_L})\) denotes the unique quadratic subfield of \(N\), is proved to be of the form \(3^eb\) with \(e\in \lbrace 1,2\rbrace\), which admits statements concerning primitive ambiguous principal ideals, lattice minima, and independent units in \(L\). The number \(m\) of non-isomorphic cubic fields \(L_1,\ldots ,L_m\) sharing a common discriminant \(d_{L_i}=d_L\) with \(L\) is determined.

On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms

Let G be an additive finite abelian group and \(\Gamma \subset {\rm End} (G)\) be a subset of the endomorphism group of G. A sequence \(S = g_1 \cdot \ldots \cdot g_{\ell}\) over G is a (\(\Gamma\)-)weighted zero-sum sequence if there are \(\gamma_1, \ldots, \gamma_{\ell} \in \Gamma\) such that \(\gamma_1 (g_1) + \ldots + \gamma_{\ell} (g_{\ell})=0\). We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group \(\Gamma\)) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences.

Prime Spectrum of the Ring of Adeles of a Number Field

Much is known about the adele ring of an algebraic number field from the perspective of harmonic analysis and class field theory. However, its ring-theoretical aspects are often ignored. Here, we present a description of the prime spectrum of this ring and study some of the algebraic and topological properties of these prime ideals. We also study how they behave under separable extensions of the base field and give an indication of how this study can be applied in adele rings not of number fields.

On second order linear divisibility sequences over algebraic number fields

On second order linear divisibility sequences over algebraic number fields

On a generalization of Menon–Sury identity to number fields involving a Dirichlet character

For every positive integer n, Sita Ramaiah’s identity states that $$\begin{aligned}&\sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} \gcd (a_1+a_2-1,n) = \phi _2(n)\sigma _0(n) \\&\quad \text { where } \; \phi _2(n)= \sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} 1, \end{aligned}$$where \((\mathbb {Z}/n\mathbb {Z})^*\) is the multiplicative group of units of the ring \(\mathbb {Z}/n\mathbb {Z}\) and \(\sigma _s(n) = \displaystyle \sum \nolimits _{d\mid n}d^s\). This identity can also be viewed as a generalization of Menon’s identity. In this article, we generalize this identity to an algebraic number field K involving a Dirichlet character \(\chi \). Our result is a further generalization of recent results in Ji and Wang (Ramanujan J 53:585–594, 2020) and Sury (Rend Circ Mat Palermo 58:99–108, 2009).

On Finite Subgroups in the General Linear Groups over an Algebraic Number Field

As is well-known, there are only finitely many isomorphic classes of finite subgroups in a given general linear group over the field of rational numbers. This result can be generalized to any algebraic number field. While the case of field of rational numbers is relatively well-studied, we still do not know much for general algebraic number field cases. In this article, we discuss the finiteness of isomorphic classes of finite subgroups of general linear groups over an algebraic number field. We give a method to calculate a multiplicative bound for the orders of finite subgroups and to classify finite cyclic subgroups.

Cyclotomic Construction of Sparse Code Multiple Access With Improved Diversity

A number of studies focus on sparse code multiple access (SCMA) to increase the spectral efficiency and support the massive connectivity required for future wireless communications networks. This work reports an improved design for SCMA based on algebraic number field theory and cyclotomic construction of a lattice. The proposed constellation improves the diversity of the SCMA system, which supports the probability of decoding error. We show that the proposed constellation in the uplink transmission of SCMA supports the spectral efficiency by 1 dB gain in SNR when the error probability of SCMA is 0.001 as compared to the proposals in the literature.