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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

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 $x^6+ax^3+b\in F[x]$.

Acknowledgment

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.

Citation

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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. https://doi.org/10.35834/2021/3302163

Information

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

Digital Object Identifier: 10.35834/2021/3302163

Subjects:
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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