[Math] units of a ring of integers

Question

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!

Answer 1

While I'm sure you know this, it's probably worth mentioning that most of the time, the real root of a given cubic polynomial with one real root won't be a unit at all, let alone fundamental. So for the rest of the discussion let's assume that $P\,$ has constant term $\pm 1$.

There are at least a few results in this direction, all stemming from the fact that by the cubic formula, computations in cubic fields (e.g., roots, discriminants, etc.) can be made very explicit. An excellent summary of these is Found in Frolich + Taylor's chapter on cubic and sextic fields in their Algebraic Number Theory, in which they explicitly address cubic number fields with precisely one real embedding.

I'll just summarize one super-handy lemma due to Artin and an application due to Ishida: Let $K$ be a cubic field with precisely one real embedding, and of discriminant $\Delta$.

If $u>1$ is a unit of $K$ such that $4u^{3/2}+24<|\Delta|$, then $u$ is a fundamental unit.

Back to polynomials, which was the original question: Here's a nice class of polynomials where you can check, very explicitly via the cubic formula, that the one real root of the polynomial satisfies the above bound:

Theorem (Ishida, p.202 in Frolich-Taylor): Suppose $\ell\geq 2$ has the property that $4\ell^3+27$ is square-free, and let $v$ denote the unique real root of $X^3+\ell X-1$. Then $v^{-1}$ is a fundamental unit of $\mathbb{Q}(v)$.

The calculations get somewehat trickier for arbitrary cubic polynomials, but it seems plausible that you could find a similar result for the natural generalization of your class of "nice" polynomials.