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ON GALOIS GROUPS OF POWER COMPOSITIONAL NONIC POLYNOMIALS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 10 February 2025, pp. 1-15
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- Article
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Let
$g(x)=x^3+ax^2+bx+c$ and 
$f(x)=g(x^3)$ be irreducible polynomials with rational coefficients, and let 
$ {\mathrm{Gal}}(f)$ be the Galois group of 
$f(x)$ over 
$\mathbb {Q}$. We show 
$ {\mathrm{Gal}}(f)$ is one of 11 possible transitive subgroups of 
$S_9$, defined up to conjugacy; we use 
$ {\mathrm{Disc}}(f)$, 
$ {\mathrm{Disc}}(g)$ and two additional low-degree resolvent polynomials to identify 
$ {\mathrm{Gal}}(f)$. We further show how our method can be used for determining one-parameter families for a given group. Also included is a related algorithm that, given a field 
$L/\mathbb {Q}$, determines when L can be defined by an irreducible polynomial of the form 
$g(x^3)$ and constructs such a polynomial when it exists.
