%0 Journal Article
%T Irreducible Polynomials in Ζ[x] That Are Reducible Modulo All Primes
%A Shiv Gupta
%J Open Journal of Discrete Mathematics
%P 52-61
%@ 2161-7643
%D 2019
%I Scientific Research Publishing
%R 10.4236/ojdm.2019.92006
%X The polynomial x<sup>4</sup>+1 <span><span style=\"font-family:Verdana;\">is
irreducible in <span style=\"white-space:nowrap;\">&amp;#918;</span></span><span style=\"white-space:nowrap;font-family:Verdana;\">[x]</span><span style=\"font-family:Verdana;\"></span></span><span></span><span><span style=\"font-family:Verdana;\"> but is
locally reducible, that is, it factors modulo </span><i><span style=\"font-family:Verdana;\">p</span></i><span style=\"font-family:Verdana;\"> for all primes </span><i><span style=\"font-family:Verdana;\">p</span></i><span style=\"font-family:Verdana;\">. In
this paper we investigate this phenomenon and prove that</span></span><span> </span><span><span style=\"font-family:Verdana;\">for any composite natural number </span><i><span style=\"font-family:Verdana;\">N</span></i><span style=\"font-family:Verdana;\"> there are monic irreducible polynomials in <span style=\"white-space:nowrap;\">&amp;#918;</span></span><span style=\"white-space:nowrap;font-family:Verdana;\">[x]</span><span style=\"font-family:Verdana;\"></span><span style=\"font-family:Verdana;\"></span></span><span></span><span style=\"font-family:Verdana;\"> which are
reducible modulo every prime.</span>
%K Irreducible Polynomial
%K Reducible Polynomial
%K Galois Theory
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=91574