Let $a,b,c,d$ be rational numbers such that $a^3c+b^3-6abc-abd+9c^2+3cd+d^2=0$ and $c \neq -c -(a^3-6ab+3d)/9$. Is it true, that one of the polynomials $p(t) = t^3-at^2+bt-c$ or $p^*(t) = t^3-at^2+bt+c+(a^3-6ab+3d)/9$ is irreducible over the rationals? I have done some computer experiments with SAGE Math, and it seems that the answer is yes, but I can't come up with a proof.
A question about irreducible polynomials on a variety
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abstract-algebra
irreducible-polynomials
affine-varieties
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0Now posted to MO, http://mathoverflow.net/questions/260743/a-question-about-irreducible-polynomials-on-a-variety – 2017-01-28