finding roots of polynomial in MAGMA or GAP

Question

Please hint me. I want to find the roots of $x^3+(1-2n)x^2-(1+4n)x+8n^2-10n-1$ with Magma or GAP. In Magma:

$P<x> := PolynomialRing(Rationals())$;

$P<n>:= PolynomialRing(Rationals())$;

$f:=x^3+(1-2n)x^2-(1+4n)x+8n^2-10n-1$;

Roots(f);

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but this polynomial has roots for $n\in\mathbb{N}$.

Answer 1

I don't think you would be able to do better than the generic formulas for roots of a degree polynomial (e.g. Wolfram $\alpha$ will give you these expressions), and these roots are not conveniently represented in the systems you refer to:

The Galois group of the polynomial in $Q(n)[x]$ is $S_3$. So the roots are not cyclotomic, but only lie in a nonabelian extension.