Glasnik Matematicki, Vol. 45, No.1 (2010), 31-41.

ON THE REDUCIBILITY OF CERTAIN QUADRINOMIALS

Jonas Jankauskas

Abstract.   In 2007 West Coast Number Theory conference Walsh asked to determine all irreducible polynomials of the form P(x) = xⁱ + x^j + x^k + 4 with integer exponents i > j > k > 0, such that for some positive integer l the polynomial P(x^l) is reducible in Z[x]. In this paper we prove that such polynomials are quadrinomials x^4m + x^3m + x^2m + 4, where m is an odd positive integer. In addition, Walsh asked for the examples of reducible quadrinomials xⁱ + x^j + x^k + n, n > 4 with no linear or quadratic factors. We compute the examples of reducible polynomials of the form above with non-trivial factors and negative coefficient n.

2000 Mathematics Subject Classification.   12E05.

Key words and phrases.   Reducibility, quadrinomials.

DOI: 10.3336/gm.45.1.03

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