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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!

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2 Answers 2

up vote 5 down vote accepted

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.

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Thanks a lot for these results. Now I have some readings for the coming weeks! –  Prénom Nom Jan 20 '12 at 17:39
1  
Sure thing. Thanks for having a name that made me laugh. –  Cam McLeman Jan 20 '12 at 18:08
    
+1 Good explanation/background. That lemma of Artin's is short and sweet. Incidentally, the general case for cubics is known. See my answer. –  Mark S. Nov 10 '13 at 18:09

The answer to the original question was fully answered back around 1930 by T. Nagell as Satz XXII of Zur Theorie der kubischen Irrationalitäten. It was reproved by Louboutin in 2005: see Louboutin's The class-number one problem for some real cubic number fields with negative discriminants.

First note that we should only be worried about cubic polynomials over $\mathbb Z$ which are $\mathbb Q$-irreducible with negative discriminant (so only one real root) and have constant term $\pm1$. We may assume that $r>1$ and the constant term is $-1$, because by substituting $1/x$ or $-1/x$ or $-x$ for $x$, symmetry covers the other cases.

Under those assumptions, $r$ is a fundamental unit except when it's a square of the fundamental unit for $x^3-m^2x^2-2mx-1$ for $m\ge1$ ($m=2$ is the case you found) or in 8 other exceptional cases: $x^3+ax^2+bx-1$ for $(a,b)=(-2,1),(-3,2),(-2,-3),(-5,4),(-12,-7),(-4,3),(-6,-5),(-7,5)$.


As an aside, this raises the question: what about the positive discriminant case, when there are three real roots? Then when is a root a fundamental unit? This was answered by S.B. Mulay and myself around 2010 in The positive discriminant case of Nagell's theorem for certain cubic orders, although a nicer proof was discovered approximately simultaneously by Louboutin, which can be found in On the fundamental units of a totally real cubic order generated by a unit. The result is similar: under the assumption (safe by symmetry) that the largest root is greater than 1 and the other two are smaller than 1 in absolute value, the only exceptions are of the form $x^3-m^2x^2+2mx-1$ for $m\ge3$ and the single sporadic case of $x^3-6x^2+5x-1$.

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