Let $\kappa(x)=x^3+a_2x^2+z^jx+a_0$ for natural number $j$ , nonzero integers $a_2$ and $a_0$ , and that rational number $z$ satisfies the equation $z^j=\frac{a_0}{a_2}$ . What is the probability that $\kappa$ has irrational roots?

9/5/2019: correction made, thanks to comment below by: Joe Mansley United Kingdom, 18 years old About: I'm weird.

$\frac{2}{3}$
$\frac{1}{4}$
$\frac{1}{2}$
$\frac{1}{3}$

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G-filtered Polycules