The GCD property and irreduciable quadratic polynomials
نویسندگان
چکیده
منابع مشابه
The Gcd Property and Irreducible Quadratic Polynomials
The proof of the following theorem is presented: If D is, respectively, a Krull domain, a Dedekind domain, or a PrGfer domain, then D is correspondingly a UFD, a PID, or a Bezout domain if and only if every irreducible quadratic polynomial in D[X] is a prime element.
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where θ = 1 + √ d 2 if d mod 4 = 1 and θ = √ d if d mod 4 = 2, 3. The purpose of this paper is to compute the extended gcd of to quadratic integers in ring Z[θ]. We assume throughout that Z[θ] is principal ideal ring, but not necessarily an euclidean ring. If [a, b+ cθ] is the module {ax+(b+ cθ)y, x, y ∈ Z}, it can be shown [3] that I is an ideal of Z[θ] if and only if I = [a, b+ cθ]; where a, ...
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Some widely known techniques can be used to factorise univariate polynomials over the domain of integers. However, finding algorithms which factorise univariate and multivariate polynomials over Z and other domains is a little trickier. Several factorisation algorithms first need GCDs of the polynomials. Computing GCDs of polynomials is also necessary for adding rational functions. Both problem...
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Given two polynomials F and G in R[x1, . . . , xn], we are going to find the nontrivial approximate GCD C and polynomials F , G ∈ R[x1, . . . , xn] such that ||F − CF ′|| < and ||G − CG′|| < , for some and some well defined norm. Many papers 1,2,3,5,8,10,11,13,15 have already discussed the problem in the case n = 1. Few of them 2,10,11 mentioned the case n > 1. Approximate GCD computation of un...
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− The problem of finding the greatest common divisor (GCD) of univariate polynomials appears in many engineering fields. Despite its formulation is well-known, it is an ill-posed problem that entails numerous difficulties when the coefficients of the polynomials are not known with total accuracy, as, for example, when they come from measurement data. In this work we propose a novel GCD estimati...
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ژورنال
عنوان ژورنال: International Journal of Mathematics and Mathematical Sciences
سال: 1986
ISSN: 0161-1712,1687-0425
DOI: 10.1155/s0161171286000893