I am currently studying algebraic number theory and I have discovered recently Dirichlet's unit theorem. Since some things are quite too abstract for me, I try to understand it with some examples.
I have considered the polynomial $P(X)=X^3-4X^2-4X-1$. It has 1 real root (call it $r$) and 2 conjugate complex roots, so the set of unity is cyclic. I thought that it was generated by r, and I was quite surprised to see that $1+2r$ is another unit. After some computations I have found that actually the generator of the group of unity is $1+\frac{2}{r}$, which is the square of $r$.
So, I wonder whether there exists some conditions for a cubic polynomial $P$ with 1 real root and 2 complex conjugates roots such that the real root is a generator of the group of unity (in my example the "nice" polynomial is $X^3-2X^2-1$). Thanks by advance for any hint or reference!