The discriminant and Stickelberger's Theorem

Question

Let $\theta$ be a root of the polynomial $x^3-x+2$, which is irreducible. Consider the basis $\{1,\theta,\theta^2\}$, which is clearly a rational basis for the integer ring in $Q(\theta)$. Now, I'm asked to use Stickelberger's Theorem to show that this basis is also an integral basis. However, I've computed the discriminant for this basis which is $-104$, at which I'm a little lost since it is not squarefree. Stickelberger's Theorem states that the discriminant should be equal to either 0 or 1 mod 4, if the bases is integral, but no information is given about the converse.

Answer 1

We note that for any other basis, we must have $-104=c^2\Delta$, whence since $c^2$ is rational we can only have $c=1,4$. We verify that $\Delta \neq -26$ since $-26\neq 0,1$ mod $4$, hence $c=1$ and our original basis is integral.