Class number of $\Bbb Q(a)$ with $a^3-a+1 = 0$ is $1$

Question

Show that class number of $\Bbb Q(a)$ with $a^3-a+1 = 0$ is $1$.

Any hints? Suggested theorems? I don't want a solution.

Mistaken Attempt:

Apply Kummer-Dedekind to $x^3-x+1 \pmod p$. For just p=2 , we get $x^2(x-1)(x+1)+1 \pmod 2$ so it doesn't factor. Similarly, for larger primes we get $x^2(x-1)(x+1)+1 \pmod p$. Thus, we only get principal ideals and so the ideal class group contains only the trivial i.e. class number is 1.

Answer 1

The Minkowski bound for $k=\mathbb{Q}[x]/(x^3-x+1)$ is $M_k=\frac{8}{9\pi}\sqrt{23}\approx 1.4$.