A note on Dedekind Criterion

2020 ◽  
pp. 2150066
Author(s):  
Anuj Jakhar ◽  
Sudesh K. Khanduja
Keyword(s):  
Prime Number ◽  
Minimal Polynomial ◽  
Algebraic Integer ◽  
Algebraic Number Field ◽  
Index Formula ◽  
Valuation Theory ◽  
Valuation Rings ◽  
Number Formula ◽  
Ring Of Algebraic Integers ◽  
Polynomial Formula

Let [Formula: see text] be an algebraic number field with [Formula: see text] an algebraic integer having minimal polynomial [Formula: see text] over the field [Formula: see text] of rational numbers and [Formula: see text] be the ring of algebraic integers of [Formula: see text]. For a fixed prime number [Formula: see text], let [Formula: see text] be the factorization of [Formula: see text] modulo [Formula: see text] as a product of powers of distinct irreducible polynomials over [Formula: see text] with [Formula: see text] monic. In 1878, Dedekind proved a significant result known as Dedekind Criterion which says that the prime number [Formula: see text] does not divide the index [Formula: see text] if and only if [Formula: see text] is coprime with [Formula: see text] where [Formula: see text]. This criterion has been widely used and generalized. In this paper, a simple proof of Generalized Dedekind Criterion [S. K. Khanduja and M. Kumar, On Dedekind criterion and simple extensions of valuation rings, Comm. Algebra 38 (2010) 684–696] using elementary valuation theory is given.

ON A THEOREM OF DEDEKIND

2008 ◽  
Vol 04 (06) ◽  
pp. 1019-1025 ◽  
Author(s):  
SUDESH K. KHANDUJA ◽  
MUNISH KUMAR
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Minimal Polynomial ◽  
Algebraic Number Field ◽  
Rational Numbers ◽  
Prime Ideals ◽  
Algebraic Integers ◽  
Dedekind Domains ◽  
Modulo P ◽  
Polynomial Formula

Let K = ℚ(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let [Formula: see text] be the factorization of the polynomial [Formula: see text] obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over ℤ/pℤ. Dedekind proved that if p does not divide [AK : ℤ[θ]], then the factorization of pAK as a product of powers of distinct prime ideals is given by [Formula: see text], with 𝔭i = pAK + gi(θ)AK, and residual degree [Formula: see text]. In this paper, we prove that if the factorization of a rational prime p in AK satisfies the above-mentioned three properties, then p does not divide [AK:ℤ[θ]]. Indeed the analogue of the converse is proved for general Dedekind domains. The method of proof leads to a generalization of one more result of Dedekind which characterizes all rational primes p dividing the index of K.

Algebraic extensions in nonstandard models and Hilbert's irreducibility theorem

1988 ◽  
Vol 53 (2) ◽  
pp. 470-480 ◽  
Author(s):  
Masahiro Yasumoto
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Algebraic Extension ◽  
Algebraic Integers ◽  
Nonstandard Models ◽  
Ring Of Algebraic Integers

LetKbe an algebraic number field andIKthe ring of algebraic integers inK. *Kand *IKdenote enlargements ofKandIKrespectively. LetxЄ *K–K. In this paper, we are concerned with algebraic extensions ofK(x)within *K. For eachxЄ *K–Kand each natural numberd, YK(x,d)is defined to be the number of algebraic extensions ofK(x)of degreedwithin *K.xЄ *K–Kis called a Hilbertian element ifYK(x,d)= 0 for alldЄ N,d> 1; in other words,K(x)has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural numberω, 2ωPωand 2ω(ω3+ 1) are Hilbertian elements in*Q, where pωis theωth prime number.

On Gauss sums over dedekind domains

2021 ◽  
pp. 1-17
Author(s):  
Zhiyong Zheng ◽  
Man Chen ◽  
Jie Xu
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Integer ◽  
Algebraic Number Field ◽  
Gauss Sums ◽  
Algebraic Integers ◽  
Dedekind Domains ◽  
Finite Norm ◽  
Integer Ring ◽  
Ring Of Algebraic Integers

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

scholarly journals On Greenberg’s generalized conjecture for imaginary quartic fields

2020 ◽  
pp. 1-11
Author(s):  
Naoya Takahashi
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Group Ring ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Prime Numbers ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Number Formula

For an algebraic number field [Formula: see text] and a prime number [Formula: see text], let [Formula: see text] be the maximal multiple [Formula: see text]-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-[Formula: see text] extension of [Formula: see text] is pseudo-null over the completed group ring [Formula: see text]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.

scholarly journals p-Torsion points on elliptic curves defined over quadratic fields

1984 ◽  
Vol 96 ◽  
pp. 139-165 ◽  
Author(s):  
Fumiyuki Momose
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Quadratic Fields ◽  
Finite Degree ◽  
Torsion Points

Let p be a prime number and k an algebraic number field of finite degree d. Manin [14] showed that there exists an integer n = n(k,p) (≧0) which satisfies the condition

scholarly journals Galois Group of the Maximal Abelian Extension over an Algebraic Number Field

1957 ◽  
Vol 12 ◽  
pp. 177-189 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Field Theory ◽  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Finite Degree ◽  
Class Field ◽  
Continuous Character

The aim of the present work is to determine the Galois group of the maximal abelian extension ΩA over an algebraic number field Ω of finite degree, which we fix once for all.Let Z be a continuous character of the Galois group of ΩA/Ω. Then, by class field theory, the character Z is also regarded as a character of the idele group of Ω. We call such Z character of Ω. For our purpose, it suffices to determine the group Xl of the characters of Ω whose orders are powers of a prime number l.

On a conjecture about the quasirecognition by prime graph of some finite simple groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950070
Author(s):  
Ali Mahmoudifar
Keyword(s):  
Computer Program ◽  
Natural Number ◽  
Simple Group ◽  
Prime Number ◽  
Prime Power ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Number Formula ◽  
Finite Linear

It is proved that some finite simple groups are quasirecognizable by prime graph. In [A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups [Formula: see text] and [Formula: see text], J. Algebra Appl. 14(1) (2015) 12pp], the authors proved that if [Formula: see text] is a prime number and [Formula: see text], then there exists a natural number [Formula: see text] such that for all [Formula: see text], the simple group [Formula: see text] (where [Formula: see text] is a linear or unitary simple group) is quasirecognizable by prime graph. Also[Formula: see text] in that paper[Formula: see text] the author posed the following conjecture: Conjecture. For every prime power [Formula: see text] there exists a natural number [Formula: see text] such that for all [Formula: see text] the simple group [Formula: see text] is quasirecognizable by prime graph. In this paper [Formula: see text] as the main theorem we prove that if [Formula: see text] is a prime power and satisfies some especial conditions [Formula: see text] then there exists a number [Formula: see text] associated to [Formula: see text] such that for all [Formula: see text] the finite linear simple group [Formula: see text] is quasirecognizable by prime graph. Finally [Formula: see text] by a calculation via a computer program [Formula: see text] we conclude that the above conjecture is valid for the simple group [Formula: see text] where [Formula: see text] [Formula: see text] is an odd number and [Formula: see text].

Symbolic computation study on exact solutions to a generalized (3+1)-dimensional Kadomtsev-Petviashvili-type equation

2020 ◽  
pp. 2150116
Author(s):  
Cheng-Cheng Zhou ◽  
Xing Lü ◽  
Hai-Tao Xu
Keyword(s):  
Numerical Simulation ◽  
Exact Solutions ◽  
Prime Number ◽  
Symbolic Computation ◽  
Three Dimensional ◽  
Type Equation ◽  
Two Dimensional ◽  
Type Solution ◽  
Bilinear Operators ◽  
Number Formula

Based on the prime number [Formula: see text], a generalized (3+1)-dimensional Kadomtsev-Petviashvili (KP)-type equation is proposed, where the bilinear operators are redefined through introducing some prime number. Computerized symbolic computation provides a powerful tool to solve the generalized (3+1)-dimensional KP-type equation, and some exact solutions are obtained including lump-type solution and interaction solution. With numerical simulation, three-dimensional plots, density plots, and two-dimensional curves are given for particular choices of the involved parameters in the solutions to show the evolutionary characteristics.

Indices in a number field II

2019 ◽  
Vol 15 (01) ◽  
pp. 89-103
Author(s):  
Mohamed Ayad ◽  
Rachid Bouchenna ◽  
Omar Kihel
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Additive Group ◽  
Necessary Condition ◽  
Ring Of Integers ◽  
Number Formula

Let [Formula: see text] be a number field of degree [Formula: see text] over [Formula: see text] and [Formula: see text] its ring of integers. For a prime number [Formula: see text], we determine the types of splittings of [Formula: see text] in [Formula: see text] for which the set [Formula: see text] is of cardinality a power of [Formula: see text]. We prove that this necessary condition is also sufficient for [Formula: see text] to be a subgroup of the additive group [Formula: see text]. Consequently, we show that, in this case, the subset of [Formula: see text], [Formula: see text] is an order of the number field.

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