Let $K=\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}(\beta)$, and $\alpha=a+b\beta+c\beta^2\in K$. That is, $a,b,c \in \mathbb{Q}$. Compute $N_{K/\mathbb{Q}}(\alpha)$, $Tr_{K/\mathbb{Q}}(\alpha)$, and $\chi_{K/\mathbb{Q}}(\alpha)$ using the definitions.
"The definitions" means I need to consider the "multiplication-by-$\alpha$" map and figure out its matrix, say $M$. Then $N_{K/\mathbb{Q}}(\alpha)=det(M)$, the trace is self-explanatory, and the characteristic polynomial is $det(\alpha I-M)$.
I think I can do this computation once I figure out the basis for $K$, which I need in order to find the matrix $M$, but I don't really know how to find the basis of cubic fields. I know it will have three elements because $\beta$ has minimal polynomial $X^3-5$, but what should they be? It's not obvious to me that $\lbrace 1,\sqrt{5},\sqrt{5}^2 \rbrace$ (or anything like this) will work.
