The factorization of f(x)x n +g(x) with f(x) monic and of degree ≤2.
Joshua Harrington ; Andrew Vincent ; Daniel White
Journal de Théorie des Nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 565-578.

Dans cet article, nous étudions la factorisation des polynômes f(x)x n +g(x)∈ℤ[x] dans le cas particulier où f(x) est un polynôme quadratique unitaire avec discriminant négatif. Nous mentionnons également des résultats similaires dans le cas où f(x) est unitaire et linéaire.

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.

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DOI : https://doi.org/10.5802/jtnb.849
Classification : 11C08,  12E05,  26C10
Mots clés : polynomials, trinomials, irreducible, factorization 
@article{JTNB_2013__25_3_565_0,
     author = {Joshua Harrington and Andrew Vincent and Daniel White},
     title = {The factorization of $f(x)x^n+g(x)$ with $f(x)$ monic and of degree $\le 2$.},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {565--578},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {3},
     year = {2013},
     doi = {10.5802/jtnb.849},
     zbl = {1293.11049},
     mrnumber = {3179677},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.849/}
}
Joshua Harrington; Andrew Vincent; Daniel White. 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, Tome 25 (2013) no. 3, pp. 565-578. doi : 10.5802/jtnb.849. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.849/

[1] A. Brauer, On the irreducibility of polynomials with large third coefficient. Amer. J. Math. 70 (1948), 423–432. | MR 25490 | Zbl 0035.00902

[2] A. Brauer, On the irreducibility of polynomials with large third coefficient II. Amer. J. Math. 73 (1951), 717–720. | MR 43131 | Zbl 0042.25201

[3] J.B. Conway, Functions of One Complex Variable. New York: Springer-Verlag. | MR 503901 | Zbl 0887.30003

[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. | MR 1772434 | Zbl 0966.11046

[5] J. Harrington, On the Factorization of the Trinomials x n +cx n-1 +d. Int. J. Number Theory 08 (2012), 1513–1518. | MR 2965763 | Zbl pre06078186

[6] A. Schinzel, On the reducibility of polynomials and in particular of trinomials. Acta. Arith. 11 (1965), 1–34. | MR 180549 | Zbl 0196.31104

[7] A. Schinzel, Reducibility of polynomials and covering systems of congruences. Acta. Arith. 13 (1967), 91–101. | MR 219515 | Zbl 0171.00701