Prime ideal factorization in a number field via Newton polygons

2021 ◽  
pp. 1-15
Author(s):  
Lhoussain El Fadil
Keyword(s):  
Prime Ideal ◽  
Number Field ◽  
Newton Polygons

On the number of incongruent solutions to a quadratic congruence over algebraic integers

2019 ◽  
Vol 15 (01) ◽  
pp. 105-130
Author(s):  
Ramy F. Taki Eldin
Keyword(s):  
Prime Ideal ◽  
Number Field ◽  
Exponential Sums ◽  
Nonzero Ideal ◽  
Explicit Formulas ◽  
Algebraic Integers ◽  
Ideal Formula ◽  
Quadratic Congruence ◽  
Ring Of Algebraic Integers

Over the ring of algebraic integers [Formula: see text] of a number field [Formula: see text], the quadratic congruence [Formula: see text] modulo a nonzero ideal [Formula: see text] is considered. We prove explicit formulas for [Formula: see text] and [Formula: see text], the number of incongruent solutions [Formula: see text] and the number of incongruent solutions [Formula: see text] with [Formula: see text] coprime to [Formula: see text], respectively. If [Formula: see text] is contained in a prime ideal [Formula: see text] containing the rational prime [Formula: see text], it is assumed that [Formula: see text] is unramified over [Formula: see text]. Moreover, some interesting identities for exponential sums are proved.

A Hypergraph with Commuting Partial Laplacians

2001 ◽  
Vol 44 (4) ◽  
pp. 385-397 ◽  
Author(s):  
Cristina M. Ballantine
Keyword(s):  
Prime Ideal ◽  
Real Number ◽  
Number Field ◽  
Linear Group ◽  
General Linear Group ◽  
Commuting Operators ◽  
Affine Building ◽  
Finite Quotient ◽  
Totally Real ◽  
Real Number Field

AbstractLetFbe a totally real number field and let GLnbe the general linear group of rank n overF. Let р be a prime ideal ofFand Fрthe completion ofFwith respect to the valuation induced by р. We will consider a finite quotient of the affine building of the group GLnover the field Fр. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

On a problem in algebraic number theory

1996 ◽  
Vol 119 (2) ◽  
pp. 191-200 ◽  
Author(s):  
J. Wójcik
Keyword(s):  
Number Theory ◽  
Prime Ideal ◽  
Number Field ◽  
Algebraic Number ◽  
Multiplicative Group ◽  
Algebraic Number Field ◽  
Algebraic Number Theory ◽  
Ring Of Integers

Let K be an algebraic number field. If q is a prime ideal of the ring of integers of K and α is a number of K prime to q then Mq(α) denotes the multiplicative group generated by α modulo q. In the paper [5] there is the remark: ‘We do not know whether for all a, b, c ∈ ℚ with abc ≠ 0, |a| ≠ 1,|b| ≠ 1,|c| ≠ 1 there exist infinitely many primes q with Mq (a) = Mq (b) = Mq (c).’

scholarly journals On the unramified common divisor of discriminants of integers in a normal extension

2000 ◽  
Vol 160 ◽  
pp. 181-186
Author(s):  
Satomi Oka
Keyword(s):  
Prime Ideal ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Normal Extension ◽  
Greatest Common Divisor ◽  
Finite Degree

AbstractLet F be an algebraic number field of a finite degree, and K be a normal extension over F of a finite degree n. Let be a prime ideal of F which is unramified in K/F, be a prime ideal of K dividing such that . Denote by δ(K/F) the greatest common divisor of discriminants of integers of K with respect to K/F. Then, divides δ(K/F) if and only if

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.

scholarly journals Zeros of recurrence sequences

1991 ◽  
Vol 44 (2) ◽  
pp. 215-223 ◽  
Author(s):  
A.J. van der Poorten ◽  
H.P. Schlickewei
Keyword(s):  
Prime Ideal ◽  
Number Field ◽  
Algebraic Number ◽  
Upper Bound ◽  
Algebraic Number Field ◽  
Number Of Zeros ◽  
Characteristic Roots ◽  
Field Of Definition ◽  
Recurrence Sequences

We give an upper bound for the number of zeros of recurrence sequences defined over an algebraic number field in terms of their order, the degree of their field of definition and the number of prime ideal divisors of the characteristic roots of the sequence.

Artin's primitive root conjecture and a problem of Rohrlich

2014 ◽  
Vol 157 (1) ◽  
pp. 79-99 ◽  
Author(s):  
CHRISTOPHER AMBROSE
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Number Field ◽  
Lower Bounds ◽  
Primitive Root ◽  
Unit Group ◽  
Finitely Generated ◽  
Asymptotic Formulae

AbstractLet$\mathbb{K}$be a number field, Γ a finitely generated subgroup of$\mathbb{K}$*, for instance the unit group of$\mathbb{K}$, and κ>0. For an ideal$\mathfrak{a}$of$\mathbb{K}$let indΓ($\mathfrak{a}$]></alt-text></inline-graphic>) denote the multiplicative index of the reduction of &#x0393; in <inline-graphic name="S0305004114000206_inline3"><alt-text><![CDATA[$(\mathcal{O}_\mathbb{K}/\mathfrak{a})$* (whenever it makes sense). For a prime ideal$\mathfrak{p}$of$\mathbb{K}$and a positive integer γ let$\mathcal{I}_\gamma^\kappa(\mathfrak{p})$be the average of${ind}_{\langle a_1,\dots,a_\gamma\rangle}(\mathfrak{p})^\kappa$over all tupels$(a_1,\dots,a_\gamma)\in{(\mathcal{O}_\mathbb{K}/\mathfrak{p})^*}^\gamma$. Motivated by a problem of Rohrlich we prove, partly conditionally on fairly standard hypotheses, lower bounds for$\sum_{\mathcal{N}{\mathfrak{a}\leq x}{ind}_{\Gamma}({\mathfrak{a})^\kappa$and asymptotic formulae for$\sum_{\mathcal{N}\mathfrak{p} \leq x} {\mathcal{I}_{\gamma}^\kappa({\mathfrak{p})$.

scholarly journals The lattice of primary ideals of orders in quadratic number fields

2016 ◽  
Vol 12 (07) ◽  
pp. 2025-2040 ◽  
Author(s):  
Giulio Peruginelli ◽  
Paolo Zanardo
Keyword(s):  
Prime Ideal ◽  
Prime Number ◽  
Number Field ◽  
Number Fields ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
The Core ◽  
Number Formula ◽  
Primary Ideals

Let [Formula: see text] be an order in a quadratic number field [Formula: see text] with ring of integers [Formula: see text], such that the conductor [Formula: see text] is a prime ideal of [Formula: see text], where [Formula: see text] is a prime. We give a complete description of the [Formula: see text]-primary ideals of [Formula: see text]. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those [Formula: see text]-primary ideals not contained in [Formula: see text]. We get three different cases, according to whether the prime number [Formula: see text] is split, inert or ramified in [Formula: see text].

K-STRATUM LIMIT AND K-STRATUM INTEGRAL ON THE GENERALIZED NUMBER FIELD

1982 ◽  
Vol 2 (4) ◽  
pp. 375-388
Author(s):  
Jiwu Wang ◽  
Tai Kang
Keyword(s):  
Number Field
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