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Calculate the discriminant of the number field $K = \mathbb{Q}(\alpha)$, where $\alpha$ is root of the polynomial $x^3 - x- 1$.

any help is welcome, cheers!

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@lhf find the roots? – P. M. O. Sep 25 '12 at 0:47
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There are formulas for the discriminant of a cubic... – anon Sep 25 '12 at 0:51
up vote 9 down vote accepted

Let $f(X) = X^3 + aX + b$ be a polynomial in $\mathbb{Z}[X]$. The discriminant $d$ of $f(X)$ is $-(4a^3 + 27b^2)$. In your case $d = -23$. Since $-23$ is square free ($23$ is a prime), $1, \alpha, \alpha^2$ is an integral basis of $K$. Hence the discriminant of $K$ is $-23$.

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