Compute the decomposition of $5\mathbb{Z}_K$ as a product of prime ideals

Question

Let $K = \mathbb{Q}(\alpha)$ such that $\alpha^3 - 5\alpha + 5 = 0$.

It is easy to show that $\mathbb{Z}_K = \mathbb{Z}[\alpha]$ and that $5$ is not maximal in $\mathbb{Z}[\alpha]$. So we cannot use the usual method of trying to factorise our minimal polynomial, $x^3 - 5x + 5$, modulo $5$.

Methods I have thought about trying are: find a primitive element $\beta\in\mathbb{Z}_K$ such that $\mathbb{Z}[\beta]$ is maximal at $5$ - but I'm not really sure how to go about this.

Any help is appreciated.

Answer 1

You can write $\alpha^3 = 5(\alpha - 1)$ thus:

if $(\alpha -1)$ is a unit, we get: $(5)=(\alpha)^3$.

Now isolate $-1$ this way: $\alpha^3 -5\alpha +4 =-1$. Since $\alpha=1$ makes the left hand side $0$, then $(\alpha -1)$ divides it, then the latter is a unit.