Irreducible Polynomials in Ζ[x] That Are Reducible Modulo All Primes - Open Journal of Discrete Mathematics

OJDM > Vol.9 No.2, April 2019

Irreducible Polynomials in Ζ[x] That Are Reducible Modulo All Primes ()

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DOI: 10.4236/ojdm.2019.92006 1,146 Downloads 3,724 Views Citations

Author(s)

Shiv Gupta

Affiliation(s)

Department of Mathematics, West Chester University, West Chester, USA.

ABSTRACT

The polynomial x⁴+1 is irreducible in Ζ[x] but is locally reducible, that is, it factors modulo p for all primes p. In this paper we investigate this phenomenon and prove that for any composite natural number N there are monic irreducible polynomials in Ζ[x] which are reducible modulo every prime.

KEYWORDS

Irreducible Polynomial, Reducible Polynomial, Galois Theory

Share and Cite:

Gupta, S. (2019) Irreducible Polynomials in Ζ[x] That Are Reducible Modulo All Primes. Open Journal of Discrete Mathematics, 9, 52-61. doi: 10.4236/ojdm.2019.92006.

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