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 $x^6+ax^3+b\in F[x]$.
We thank the anonymous referees for their close reading and helpful comments, especially regarding the proofs of Proposition 2.3 and Theorem 2.4. Their feedback has greatly improved the paper's exposition.
"Field Extensions Defined by Power Compositional Polynomials." Missouri J. Math. Sci. 33 (2) 163 - 180, Novemberr 2021. https://doi.org/10.35834/2021/3302163