Reducibility of quartic polynomials and its relation to Galois groups

Abstract

This paper deals with special classes of quartic polynomials and properties pertaining to their Galois groups and reducibility over certain fields. The existence of quartic polynomials irreducible over Q but reducible over every prime field is first proven, after which criteria are established for the Galois group of polynomials with this property. By constructing classes of V4-generic polynomials and comparing them with criteria put forth in previous studies for determining polynomials with this property, it can be shown that a polynomial of the biquadratic form x4 + ax2 + b has this property if and only if it can be written as x^4 - 2(u + v)x^2 + (u -v)^2 with u, v Q such that none of u, v, or uv can be expressed as ratio of two squares, and 2(u+v),(uv)2 Z . The general form for biquadratic polynomials irreducible over Q and reducible modulo every integer n is found to have a general form similar to this one.