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GALOIS GROUPS OF RECIPROCAL SEXTIC POLYNOMIALS
Published online by Cambridge University Press: 27 February 2023
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.
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 109 , Issue 1 , February 2024 , pp. 37 - 44
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.