Let $\alpha$ be a root of the irreducible cubic polynomial $x^{3}+px+q$, $p,q\in \mathbb{Q}$. How can I compute the discriminant $\Delta(1,\alpha,\alpha^{2})$ relative to $\mathbb{Q}(\alpha)$?
How to compute a discriminant
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algebraic-number-theory
2 Answers
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If $\alpha=\alpha_1,\alpha_2, \alpha_3$ are the different roots of $f=x^3+px+q$, then \begin{align*} \Delta(1,\alpha,\alpha^2) &= \det \left( \matrix{1&\alpha_1&\alpha_1^2\\1&\alpha_2 &\alpha_2^2\\ 1&\alpha_3&\alpha_3^2} \right)^2 \quad \text{by definition of the discriminant}\\ &= \prod_{i
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The easiest would be to not use the discriminant, but to rather draw up a variation table with the help of differentiation.