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GALOIS GROUPS OF RECIPROCAL SEXTIC POLYNOMIALS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 27 February 2023, pp. 37-44
- Print publication:
- February 2024
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- Article
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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.
Galois Groups of Even Sextic Polynomials
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- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 3 / September 2020
- Published online by Cambridge University Press:
- 12 December 2019, pp. 670-676
- Print publication:
- September 2020
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Let
$f(x)=x^{6}+ax^{4}+bx^{2}+c$ be an irreducible sextic polynomial with coefficients from a field 
$F$ of characteristic 
$\neq 2$, and let 
$g(x)=x^{3}+ax^{2}+bx+c$. We show how to identify the conjugacy class in 
$S_{6}$ of the Galois group of 
$f$ over 
$F$ using only the discriminants of 
$f$ and 
$g$ and the reducibility of a related sextic polynomial. We demonstrate that our method is useful for producing one-parameter families of even sextic polynomials with a specified Galois group.
