I am working through Neukirch's Algebraic Number Theory on my own. Exercise 6 in Section 1 (page 5) is to show that the ring $\mathbb{Z}[\sqrt{d}]$, for any squarefree rational integer $d>1$, has infinitely many units.
I know that in $\mathbb{Z}[\sqrt 2]$ there are infinitely many units, because
$(\sqrt{2} + 1)(\sqrt{2} – 1) = 1$ and then taking $n$th powers shows that $(\sqrt{2} + 1)^n$ is a unit for any $n\ge 1$.
Similarly in $\mathbb{Z}[\sqrt{3}]$, we have $(2+\sqrt{3})(2-\sqrt{3}) = 1$, and then $(2+\sqrt{3})^n$ for $n\ge 1$ is an infinite family of units.
I can find other "fundamental units" for other specific values of $d$.
But it seems I have to show that the (Pell) equation $a^2 – db^2 = \pm 1$, for any $d>1$, has an integer solution $(a, b) \ne (\pm 1, 0)$, because I know if I can find one solution, then I can get infinitely many. But from my limited knowledge of Pell's equation this is a difficult problem (using techniques such as continued fractions.)
Maybe there is a simpler nonconstructive proof that I'm missing. Any hints or suggestions?
Best Answer
I don't actually know any of the standard proofs of Dirichlet's unit theorem that Qiaochu Yuan refers to, but I think they might use Minkowski's theorem about the existence of nonzero lattice points in sufficiently large centrally symmetric convex subsets of $\mathbf R^n$. I'll give a proof for the specific question asked here, which uses Minkowski's theorem, but only in the very simple case of a parallelogram. I'll write the proof top-down, so that one sees how the theorem is used before I'll state (and prove) it.
First I recall some generalities about the ring $R=\mathbf Z[\sqrt d]$ for $d$ a positive squarefree number, to get started. The additive group of $R$ is free Abelian of rank $2$ with generators $1,\sqrt d$. The norm map $N:R\to\mathbf Z$ given by $a+b\sqrt d\mapsto a^2-db^2$ is multiplicative, and the units of $R$ are precisely the elements with norm $\pm1$. One has the non-trivial unit $-1$, but it is of finite order; the point to prove is therefore that the subgroup of positive units is non-trivial (once a positive unit${}\neq1$ is found, its powers form an infinite set of units). I will reason by contradiction, so assume that $1$ is the unique positive unit of $R$. The first step will be to show that this would imply that for any $n\in\mathbf N$, the number of positive $r\in R$ with $|N(r)|=n$ is finite, in fact at most $n^2$.
Lemma. For any $n\in\mathbf N_{>0}$, the number of principal ideals of $R$ that contain $n$ is at most $n^2$.
Proof. Since these ideals all contain $nR$, they all map to principal ideals of $R/nR$, and the mapping is injective. The number of principal ideals of $R/nR$ cannot exceed the number $n^2$ of its elements. QED
The bound given here is far from sharp, but finiteness is all we need. Under the hypothesis that $1$ is the unique positive unit of $R$, two positive elements of $R$ generate distinct principal ideals, and if $|N(r)|=n$, the ideal generated by $r$ contains $n$, so the lemma justifies our claim that the number of of such$~r$ is finite.
This means that for every $M>0$ there is some $\varepsilon_M>0$ such that for all $a,b\in\mathbf Z$ with $0\neq|a+b\sqrt d|<\varepsilon_M$ one has $|N(a+b\sqrt d)|\geq M$. We shall show that for sufficiently large $M$ this contradicts Minkowski's theorem. Note that $N(a+b\sqrt d)=(a+b\sqrt d)(a-b\sqrt d)$, so we can bound $N(a+b\sqrt d)$ above if in addition to the value $a+b\sqrt d$ we also bound its conjugate $a-b\sqrt d$. Now the linear endomorphism of $\mathbf R^2$ sending $\binom ab\mapsto\binom{a+b\sqrt d}{a-b\sqrt d}$ has determinant $-2\sqrt d$, so the conditions $|a+b\sqrt d|<x$ and $|a-b\sqrt d|<y$ define the interior of a parallelogram of area $4\frac{xy}{2\sqrt d}$, for any $x,y>0$. Minkowski's theorem says such a parallelogram will contain a nonzero lattice point whenever its area is greater than $2^2=4$.
So here is how to obtain a contradiction: take any $M>2\sqrt d$ and put $$ x_0=\min\{ z\in R \mid z>0 \land |N(z)|\leq M\}\qquad\text{and}\qquad y_0=\frac M{x_0}. $$ Then $\frac{x_0y_0}{2\sqrt d}>1$, so Minkowski's theorem ensures the existence of $a,b\in\mathbf Z$, not both $0$, with $|a+b\sqrt d|<x_0$ and $|a-b\sqrt d|<y_0$. But then on one hand $|N(a+b\sqrt d)|>M$ by the choice of $x_0$ (and the fact $N(-z)=N(z)$), but on the other hand $|N(a+b\sqrt d)|=|a+b\sqrt d||a-b\sqrt d|<x_0y_0=M$, a contradiction.
Minkowski's theorem. Any centrally symmetric convex subset $S$ of $\mathbf R^d$ of volume greater than $2^d$ contains a nonzero element of $\mathbf Z^d$.
Proof. The natural projection map $f:\mathbf R^d\to(\mathbf R/2\mathbf Z)^d$ is locally volume-preserving, so its restriction to $S$ cannot be injective since the total volume at arrival is $2^d$. If $s,s'\in S$ have $s\neq s'$ and $f(s)=f(s')$ then by central symmetry $-s'\in S$, and by convexity $\frac{s-s'}2\in S$; therefore, since $f(s-s')\in(2\mathbf Z)^d$, one has $\frac{s-s'}2\in S\cap(\mathbf Z^d\setminus\{0\})$. QED