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Tests and Constructions of Irreducible Polynomials over Finite Fields
 In Foundations of Computational Mathematics
, 1997
"... In this paper we focus on tests and constructions of irreducible polynomials over finite fields. We revisit Rabin's [1980] algorithm providing a variant of it that improves Rabin's cost estimate by a log n factor. We give a precise analysis of the probability that a random polynomial of de ..."
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Cited by 12 (4 self)
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In this paper we focus on tests and constructions of irreducible polynomials over finite fields. We revisit Rabin's [1980] algorithm providing a variant of it that improves Rabin's cost estimate by a log n factor. We give a precise analysis of the probability that a random polynomial
CONSTRUCTING IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS
"... Abstract. We describe a new method for constructing irreducible polynomials modulo a prime number p. The method mainly relies on Chebotarev’s density theorem. ..."
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Abstract. We describe a new method for constructing irreducible polynomials modulo a prime number p. The method mainly relies on Chebotarev’s density theorem.
Parallel Construction of Irreducible Polynomials
, 1991
"... Let arithmetic pseudoNC k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field Fp of depth O(log k (n + p)) and size (n + p) O(1). We show that the problem of constructing an irreducible polynomial of specified degree over Fp belongs to pseudo ..."
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Cited by 1 (0 self)
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Let arithmetic pseudoNC k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field Fp of depth O(log k (n + p)) and size (n + p) O(1). We show that the problem of constructing an irreducible polynomial of specified degree over Fp belongs to pseudo
Iterated constructions of irreducible polynomials over finite fields with linearly independent roots
, 2004
"... The paper is devoted to constructive theory of synthesis of irreducible polynomials and irreducible Npolynomials (with linearly independent roots) over finite fields. For a suitably chosen initial Npolynomial F1ðxÞAF2s x of degree n; polynomials FkðxÞAF2s x of degrees 2k1n are constructed by ite ..."
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Cited by 4 (0 self)
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The paper is devoted to constructive theory of synthesis of irreducible polynomials and irreducible Npolynomials (with linearly independent roots) over finite fields. For a suitably chosen initial Npolynomial F1ðxÞAF2s x of degree n; polynomials FkðxÞAF2s x of degrees 2k1n are constructed
Construction and Distribution Problems for Irreducible Trinomials over Finite Fields
 in Applications of Finite Fields
, 2001
"... Various aspects of the construction, distribution and factorization of polynomials over a finite field have been widely investigated in the recent literature. The present work is limited to studying certain questions on the distribution of irreducible trinomials over F 2 . All such irreducible trino ..."
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Cited by 18 (2 self)
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Various aspects of the construction, distribution and factorization of polynomials over a finite field have been widely investigated in the recent literature. The present work is limited to studying certain questions on the distribution of irreducible trinomials over F 2 . All such irreducible
Deterministic Irreducibility Testing of Polynomials over Large Finite Fields
 J. Symbolic Comput
, 1987
"... We present a sequential deterministic polynomialtime algorithm for testing dense multivariate polynomials over a large finite field for irreducibility. All previously known algorithms were of a probabilistic nature. Our deterministic solution is based on our algorithm for absolute irreducibility te ..."
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Cited by 8 (3 self)
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We present a sequential deterministic polynomialtime algorithm for testing dense multivariate polynomials over a large finite field for irreducibility. All previously known algorithms were of a probabilistic nature. Our deterministic solution is based on our algorithm for absolute irreducibility
ON IRREDUCIBLE FACTORS OF POLYNOMIALS OVER COMPLETE FIELDS
"... Abstract. Let (K, v) be a complete rank1 valued field. In this paper, we extend classical Hensel’s Lemma to residually transcendental prolongations of v to a simple transcendental extension K(x) and apply it to prove a generalization of Dedekind’s theorem regarding splitting of primes in algebraic ..."
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Cited by 1 (1 self)
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number fields. We also deduce an irreducibility criterion for polynomials over rank1 valued fields which extends already known generalizations of Schönemann Irreducibility Criterion for such fields. A refinement of Generalized Akira criterion proved in [Manuscripta Math., 134:12 (2010) 215224] is also
Specific Irreducible Polynomials with Linearly Independent Roots over Finite Fields
"... In this paper we give several families of specific irreducible polynomials with the following property: if f(x) is one of the given polynomials of degree n over a finite field F q and # is a root of it, then # # F q n is normal over every intermediate field between F q n and F q . Here by # # F q ..."
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Cited by 3 (1 self)
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In this paper we give several families of specific irreducible polynomials with the following property: if f(x) is one of the given polynomials of degree n over a finite field F q and # is a root of it, then # # F q n is normal over every intermediate field between F q n and F q . Here by # # F q
ABSOLUTE IRREDUCIBILITY OF POLYNOMIALS
"... Abstract. A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give m ..."
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many simple irreducibility criteria including the wellknown Eisenstein’s criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping certain collection of them nonzero
IRREDUCIBILITY OF POLYNOMIALS AND DIOPHANTINE EQUATIONS
"... Abstract. In [3] we showed that a polynomial over a Noetherian ring is divisible by some other polynomial by looking at the matrix formed by the coefficients of the polynomials which we called the resultant matrix. In this paper, we consider the polynomials with coefficients in a field and divisibil ..."
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and divisibility of a polynomial by a polynomial with a certain degree is equivalent to the existence of common solution to a system of Diophantine equations. As an application we construct a family of irreducible quartics over Q which are not of Eisenstein type. 1.
Results 1  10
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