Field Extensions Defined by Power Compositional Polynomials

Abstract

Suppose $L/F$ is a finite extension of fields of characteristic 0. We study when $L$ can be defined by an irreducible power compositional polynomial; that is, a polynomial of the form $f(x^k)\in F[x]$. If $L$ contains the $k$th roots of unity, then Kummer theory provides an answer. In this case, $f(x^k)$ can be constructed explicitly using automorphisms of $L/F$. If $L$ does not contain the $k$th roots of unity, we use a result of Kang to give a complete answer when $k=3$ and $[L:F]=6$, and we construct a polynomial $x^6+ax^3+b$ defining $L/F$. As an application, we give a simple method for determining the Galois group of an irreducible polynomial $x^6+ax^3+b\in F[x]$.