TITLE: Find a non-trivial unit of the ring of integers of a specific cubic field.
QUESTION [5 upvotes]: In particular, how to find a non-trivial unit of the ring of integers of $\mathbb{Q}(\alpha)$, where $\alpha$ is a root of $y^3+3y-5=0$.
Note, since the constant term is $-5$, the norm of $\alpha$ is 5, thus $\alpha$ is not a unit.
So far, I can prove that a non-trivial unit is not in the form of $a\alpha+b$, where $a,b\in\mathbb{Z}$.

REPLY [2 votes]: $$(y-1)(y^2+y+4)=(y^3+3y-5)+1$$