Let $K=\mathbb Q(\alpha)$ where $ \alpha^3 -5\alpha + 5 = 0 $. I need to show that the prime ideals above 5 are principal, and find a generator for them.
I have worked out the prime decomposition of $5\mathbb Z_k$ as $5\mathbb Z_k = (5,\alpha)^3 $ by factorising the minimal polynomial of $\alpha$ modulo 5, in particular giving that $K$ is totally ramified at $p=5$. But now i'm not sure how to show that the ideals above 5 are principal, or how it follows from my previous calculations.
From your equation, given that $\alpha$ is a solution of $x^3−5x+5=0$, it follows that $$ 5=(-\alpha^2+5)\alpha $$ so $5$ is already in the ideal generated by $\alpha$.
In other words, $(5, \alpha) = (\alpha)$ and the ideal is principal.