Let $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of $X^{3}-4 X+1$. Prove that $K$ has degree 3 over $\mathbb{Q}$, and admits three distinct embeddings in $\mathbb{R}$. Find the discriminant of $K$ and determine the ring of integers $\mathcal{O}$ of $K$. Factorise $2 \mathcal{O}$ and $3 \mathcal{O}$ into a product of prime ideals.

Using Minkowski's bound, show that $K$ has class number 1 provided all prime ideals in $\mathcal{O}$ dividing 2 and 3 are principal. Hence prove that $K$ has class number $1 .$

[You may assume that the discriminant of $X^{3}+a X+b$ is $-4 a^{3}-27 b^{2}$.]