Novemberr 2021 Field Extensions Defined by Power Compositional Polynomials
Chad Awtrey, James R. Beuerle, Hanna Noelle Griesbach
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Missouri J. Math. Sci. 33(2): 163-180 (Novemberr 2021). DOI: 10.35834/2021/3302163


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.


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Chad Awtrey. James R. Beuerle. Hanna Noelle Griesbach. "Field Extensions Defined by Power Compositional Polynomials." Missouri J. Math. Sci. 33 (2) 163 - 180, Novemberr 2021.


Published: Novemberr 2021
First available in Project Euclid: 30 November 2021

MathSciNet: MR4345237
zbMATH: 1486.12004
Digital Object Identifier: 10.35834/2021/3302163

Primary: 12F05
Secondary: 11R32 , 12-08 , 20B35

Keywords: automorphisms , Galois groups , Kummer theory , power compositional , radical extensions , roots of unity , sextic extensions

Rights: Copyright © 2021 University of Central Missouri, School of Computer Science and Mathematics


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Vol.33 • No. 2 • November 2021
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