scholarly journals $ \Omega $-result for the index of composition of an integral ideal

2021 ◽  
Vol 6 (5) ◽  
pp. 4979-4988
Author(s):  
Jing Huang ◽  
◽  
Wenguang Zhai ◽  
Deyu Zhang ◽  
Keyword(s):  
Integral Ideal

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals Integral ∨-ideals

1981 ◽  
Vol 22 (2) ◽  
pp. 167-172 ◽  
Author(s):  
David F. Anderson
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Dedekind Domain ◽  
Integral Ideal ◽  
Quotient Field ◽  
Fractional Ideal

Let R be an integral domain with quotient field K. A fractional ideal I of R is a ∨-ideal if I is the intersection of all the principal fractional ideals of R which contain I. If I is an integral ∨-ideal, at first one is tempted to think that I is actually just the intersection of the principal integral ideals which contain I.However, this is not true. For example, if R is a Dedekind domain, then all integral ideals are ∨-ideals. Thus a maximal ideal of R is an intersection of principal integral ideals if and only if it is actually principal. Hence, if R is a Dedekind domain, each integral ∨-ideal is an intersection of principal integral ideals precisely when R is a PID.

scholarly journals EDUCAÇÃO INTEGRAL: IDEAL ACALENTADO E CONCEITO MALTRATADO

2020 ◽  
pp. 174-200
Author(s):  
Paulo Moacir Godoy Pozzebon ◽  
Samuel Mendonça
Keyword(s):  
Integral Ideal

O ensaio objetiva discutir formas de apropriação conceitual do ideal de educação integral. A partir do método de revisão bibliográfica, contemplando referências históricas do pensamento pedagógico, a legislação educacional brasileira e a literatura educacional brasileira contemporânea, busca-se apreender em que consiste a educação integral. Os resultados evidenciam que não há homogeneidade na compreensão da educação integral, embora, paradoxalmente, exista consenso em sua busca. Além da necessidade de maior aprofundamento conceitual e contextualização histórica, as elaborações em torno da educação integral são muito divergentes e não aprofundam o seu sentido, fornecendo balizas superficiais para políticas públicas e assimilando-a a políticas de proteção social.  Por fim, o debate sobre o tema reivindica a fundamentação a partir de uma filosofia do ser humano que discuta, efetivamente, a integralidade de suas dimensões constitutivas e permita sua expressão em termos de necessidades educativas.

Relative Galois module structure of rings of integers and elliptic functions

1983 ◽  
Vol 94 (3) ◽  
pp. 389-397 ◽  
Author(s):  
M. J. Taylor
Keyword(s):  
Number Field ◽  
Finite Extension ◽  
Elliptic Functions ◽  
Complex Numbers ◽  
Module Structure ◽  
Galois Module ◽  
Integral Ideal ◽  
Ring Of Integers ◽  
Rings Of Integers ◽  
Imaginary Number

Let K be a quadratic imaginary number field with discriminant less than −4. For N either a number field or a finite extension of the p-adic field p, we let N denote the ring of integers of N. Moreover, if N is a number field then we write for the integral closure of [½] in N. For an integral ideal & of K we denote the ray classfield of K with conductor & by K(&). Once and for all we fix a choice of embedding of K into the complex numbers .

scholarly journals On a problem of Hasse and Ramachandra

2019 ◽  
Vol 17 (1) ◽  
pp. 131-140
Author(s):  
Ja Kyung Koo ◽  
Dong Hwa Shin ◽  
Dong Sung Yoon
Keyword(s):  
Positive Integer ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Integral Ideal ◽  
Weber Function ◽  
Class Field ◽  
Ray Class Field

Abstract Let K be an imaginary quadratic field, and let 𝔣 be a nontrivial integral ideal of K. Hasse and Ramachandra asked whether the ray class field of K modulo 𝔣 can be generated by a single value of the Weber function. We completely resolve this question when 𝔣 = (N) for any positive integer N excluding 2, 3, 4 and 6.

scholarly journals On the mean value of the index of composition of an integral ideal (II)

2013 ◽  
Vol 133 (4) ◽  
pp. 1086-1110
Author(s):  
Deyu Zhang ◽  
Wenguang Zhai
Keyword(s):  
Mean Value ◽  
Integral Ideal ◽  
The Mean

A FAMILY OF ASYMPTOTICALLY GOOD LATTICES HAVING A LATTICE IN EACH DIMENSION

2008 ◽  
Vol 04 (01) ◽  
pp. 147-154 ◽  
Author(s):  
J. CARMELO INTERLANDO ◽  
ANDRÉ LUIZ FLORES ◽  
TRAJANO PIRES DA NÓBREGA NETO
Keyword(s):  
Packing Density ◽  
Cyclotomic Field ◽  
Sphere Packing ◽  
Integral Ideal ◽  
The Family

A new constructive family of asymptotically good lattices with respect to sphere packing density is presented. The family has a lattice in every dimension n ≥ 1. Each lattice is obtained from a conveniently chosen integral ideal in a subfield of the cyclotomic field ℚ(ζq) where q is the smallest prime congruent to 1 modulo n.

scholarly journals On the least prime ideal and Siegel zeros

2016 ◽  
Vol 12 (08) ◽  
pp. 2201-2229 ◽  
Author(s):  
Asif Zaman
Keyword(s):  
Prime Ideal ◽  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Forms ◽  
Integral Ideal ◽  
Siegel Zero ◽  
Siegel Zeros ◽  
Special Case ◽  
Exceptional Character

Let [Formula: see text] be a number field, [Formula: see text] be an integral ideal, and [Formula: see text] be the associated narrow ray class group. Suppose [Formula: see text] possesses a real exceptional character [Formula: see text], possibly principal, with a Siegel zero [Formula: see text]. For [Formula: see text] satisfying [Formula: see text] [Formula: see text], we establish an effective [Formula: see text]-uniform Linnik-type bound with explicit exponents for the least norm of a prime ideal [Formula: see text]. A special case of this result is a bound for the least rational prime represented by certain binary quadratic forms.

Overrings of Prüfer v-multiplication domains

2016 ◽  
Vol 16 (08) ◽  
pp. 1750147 ◽  
Author(s):  
Shiqi Xing ◽  
Fanggui Wang
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
Integral Ideal ◽  
Ideal Formula ◽  
Similar Work ◽  
Star Operation ◽  
Flat Modules ◽  
Torsion Free

Let [Formula: see text] be an integral domain, [Formula: see text] and [Formula: see text] the set of fractional ideals of [Formula: see text]. Let [Formula: see text] a finitely generated ideal with [Formula: see text]. For a torsion-free [Formula: see text]-module [Formula: see text], define [Formula: see text] for some [Formula: see text]. Call [Formula: see text] a [Formula: see text]-module if [Formula: see text]. On [Formula: see text], the function [Formula: see text] is a star-operation of finite character. An integral ideal [Formula: see text] maximal with respect to being a proper [Formula: see text]-ideal is a prime ideal called a maximal [Formula: see text]-ideal. A torsion-free [Formula: see text]-module [Formula: see text] is called [Formula: see text]-flat, if [Formula: see text] is a flat [Formula: see text]-module for each [Formula: see text], the set of maximal [Formula: see text]-ideals of [Formula: see text]. [Formula: see text] is called a Prüfer [Formula: see text]-multiplication domain (P[Formula: see text]MD), if [Formula: see text] is a valuation ring for each [Formula: see text]. We characterize [Formula: see text]-flat modules in a manner similar to the characterization of flat modules, study them when they are rings [Formula: see text] with [Formula: see text] and characterize P[Formula: see text]MDs using them and compare our work with similar work in the literature.

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