TY - JOUR
AU - Harrington, Joshua
AU - Jones, Lenny
PY - 2017/05/27
Y2 - 2022/01/17
TI - The irreducibility of power compositional sextic polynomials and their Galois groups
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 120
IS - 2
SE - Articles
DO - 10.7146/math.scand.a-25850
UR - https://www.mscand.dk/article/view/25850
SP - 181-194
AB - <p>We define a <em>power compositional sextic polynomial</em> to be a monic sextic polynomial $f(x):=h(x^d)\in \mathbb{Z} [x]$, where $h(x)$ is an irreducible quadratic or cubic polynomial, and $d=3$ or $d=2$, respectively. In this article, we use a theorem of Capelli to give necessary and sufficient conditions for the reducibility of $f(x)$, and also a description of the factorization of $f(x)$ into irreducibles when $f(x)$ is reducible. In certain situations, when $f(x)$ is irreducible, we also give a simple algorithm to determine the Galois group of $f(x)$ without the calculation of resolvents. The algorithm requires only the use of the Rational Root Test and the calculation of a single discriminant. In addition, in each of these situations, we give infinite families of polynomials having the possible Galois groups.</p>
ER -