Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
What have you tried? –  lhf Sep 25 '12 at 0:45
@lhf find the roots? –  P. M. O. Sep 25 '12 at 0:47
There are formulas for the discriminant of a cubic... –  anon Sep 25 '12 at 0:51

1 Answer 1

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 $-23$.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.