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GALOIS GROUPS OF RECIPROCAL SEXTIC POLYNOMIALS

Part of: Field extensions Computational number theory

Published online by Cambridge University Press:  27 February 2023

CHAD AWTREY Open the ORCID record for CHAD AWTREY [Opens in a new window]  and
ALLY LEE
CHAD AWTREY*
Affiliation:
Department of Mathematics and Computer Science, Samford University, 800 Lakeshore Drive, Birmingham, AL 35229, USA
ALLY LEE
Affiliation:
Department of Mathematics and Computer Science, Samford University, 800 Lakeshore Drive, Birmingham, AL 35229, USA e-mail: alee11@samford.edu
*
e-mail: cawtrey@samford.edu
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Abstract

Let F be a subfield of the complex numbers and $f(x)=x^6+ax^5+bx^4+cx^3+bx^2+ax+1 \in F[x]$ an irreducible polynomial. We give an elementary characterisation of the Galois group of $f(x)$ as a transitive subgroup of $S_6$. The method involves determining whether three expressions involving a, b and c are perfect squares in F and whether a related quartic polynomial has a linear factor. As an application, we produce one-parameter families of reciprocal sextic polynomials with a specified Galois group.

Keywords

reciprocal polynomialssexticGalois group

MSC classification

Primary: 12F10: Separable extensions, Galois theory
Secondary: 11Y40: Algebraic number theory computations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 1 , February 2024 , pp. 37 - 44
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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