The factorization of f ( x ) x n + g ( x ) with f ( x ) monic and of degree ≤ 2 .

Joshua Harrington[1]; Andrew Vincent[2]; Daniel White[1]

  • [1] Department of Mathematics University of South Carolina Columbia, SC 29208
  • [2] Department of Mathematics University of South Carolina Columbia, SC, 29208

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 3, page 565-578
  • ISSN: 1246-7405

Abstract

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In this paper we investigate the factorization of the polynomials f ( x ) x n + g ( x ) ∈ ℤ [ x ] in the special case where f ( x ) is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that f ( x ) is monic and linear.

How to cite

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Harrington, Joshua, Vincent, Andrew, and White, Daniel. "The factorization of $f(x)x^n+g(x)$ with $f(x)$ monic and of degree $\le 2$.." Journal de Théorie des Nombres de Bordeaux 25.3 (2013): 565-578. <http://eudml.org/doc/275722>.

@article{Harrington2013,
abstract = {In this paper we investigate the factorization of the polynomials $f(x)x^n+g(x)\in \mathbb\{Z\}[x]$ in the special case where $f(x)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f(x)$ is monic and linear.},
affiliation = {Department of Mathematics University of South Carolina Columbia, SC 29208; Department of Mathematics University of South Carolina Columbia, SC, 29208; Department of Mathematics University of South Carolina Columbia, SC 29208},
author = {Harrington, Joshua, Vincent, Andrew, White, Daniel},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {polynomials; trinomials; irreducible; factorization; irreducibility criteria; factorization of polynomials; quadratic polynomial},
language = {eng},
month = {11},
number = {3},
pages = {565-578},
publisher = {Société Arithmétique de Bordeaux},
title = {The factorization of $f(x)x^n+g(x)$ with $f(x)$ monic and of degree $\le 2$.},
url = {http://eudml.org/doc/275722},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Harrington, Joshua
AU - Vincent, Andrew
AU - White, Daniel
TI - The factorization of $f(x)x^n+g(x)$ with $f(x)$ monic and of degree $\le 2$.
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/11//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 3
SP - 565
EP - 578
AB - In this paper we investigate the factorization of the polynomials $f(x)x^n+g(x)\in \mathbb{Z}[x]$ in the special case where $f(x)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f(x)$ is monic and linear.
LA - eng
KW - polynomials; trinomials; irreducible; factorization; irreducibility criteria; factorization of polynomials; quadratic polynomial
UR - http://eudml.org/doc/275722
ER -

References

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  1. A. Brauer, On the irreducibility of polynomials with large third coefficient. Amer. J. Math. 70 (1948), 423–432. Zbl0035.00902MR25490
  2. A. Brauer, On the irreducibility of polynomials with large third coefficient II. Amer. J. Math. 73 (1951), 717–720. Zbl0042.25201MR43131
  3. J.B. Conway, Functions of One Complex Variable. New York: Springer-Verlag. Zbl0887.30003MR503901
  4. M. Filaseta, K. Ford, S. Konyagin, On an irreducibility theorem of A. Schinzel associated with covering of the integers. Illinois J. Math. 44(3) (2000), 633–643. Zbl0966.11046MR1772434
  5. J. Harrington, On the Factorization of the Trinomials x n + c x n - 1 + d . Int. J. Number Theory 08 (2012), 1513–1518. Zbl1293.12003MR2965763
  6. A. Schinzel, On the reducibility of polynomials and in particular of trinomials. Acta. Arith. 11 (1965), 1–34. Zbl0196.31104MR180549
  7. A. Schinzel, Reducibility of polynomials and covering systems of congruences. Acta. Arith. 13 (1967), 91–101. Zbl0171.00701MR219515

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