Finding the ring of integers of a cubic extension of $\mathbb{Q}$

Question

Let $f(x)=x^3-9x-6$ with $a\in \mathbb{C}$ a root of $f(x)$. I need to show that the ring of integers of $L=\mathbb{Q}(a)$ is $O_L=\mathbb{Z}[a]$. I computed the discriminant of $f(x)$ to be 1944. Even by Stickelberger's theorem, I narrowed down $(O_L:\mathbb{Z}[a])$ to 1 or 3. I need to show it must be 1. Any ideas how to proceed? Thanks.

Answer 1

Since $f$ is Eisenstein at $p = 3$, we can apply the following result, Theorem 2.3 of Keith Conrad's Totally Ramified Primes and Eisenstein Polynomials.

Theorem. Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of an Eisenstein polynomial at $p$. Then $p \nmid [\mathcal{O}_K : \mathbb{Z}[\alpha]]$.