Factorisation of Quintic x^5 - x + n into Irreducible Quadratic and Irreducible Cubic

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Theorem

The quintic $x^5 - x + n$ can be factorized into the product of an irreducible quadratic and an an irreducible cubic if and only if $n$ is in the set:

$\set {\pm 15, \pm 22 \, 440, \pm 2 \, 759 \, 640}$


Proof

We have that:

\(\ds x^5 - x \pm 15\) \(=\) \(\ds \paren {x^2 \pm x + 3} \paren {x^3 \mp x^2 \mp 2 x \pm 5}\)
\(\ds x^5 - x \pm 22440\) \(=\) \(\ds \paren {x^2 \mp 12 x + 55} \paren {x^3 \pm 12 x^2 + 89 x \pm 408}\)
\(\ds x^5 - x \pm 2 \, 759 \, 640\) \(=\) \(\ds \paren {x^2 \pm 12 x + 377} \paren {x^3 \mp 12 x^2 - 233 x \pm 7320}\)


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Sources

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