Dirichlet unit theorem
In the ordinary integers, the only numbers with a multiplicative inverse that is still an integer are +1 and -1 — a tiny, boring unit group. In a general ring of integers the units can be far richer: there can be infinitely many, like the fundamental solution to a Pell equation that you can raise to any power. Dirichlet's theorem says exactly how rich, in terms of the shape of the number field.
Let K be a number field with r_1 real embeddings and r_2 pairs of complex embeddings, so [K : Q] = r_1 + 2 r_2. The theorem states that the unit group of O_K is isomorphic to (the finite cyclic group of roots of unity in K) times Z^{r_1 + r_2 - 1}. In words: modulo roots of unity, the units form a free abelian group of rank r = r_1 + r_2 - 1.
The proof embeds the units via their logarithms into a hyperplane in R^{r_1 + r_2} and shows their image is a full-rank lattice — the unit group is, up to torsion, a lattice. The covolume of that lattice is the regulator, a real number measuring the “density” of units, which appears alongside the class number in the analytic class number formula.
For the real quadratic field Q(sqrt(2)), one has r_1 = 2, r_2 = 0, so the rank is 2 + 0 - 1 = 1. The units are plus or minus (1 + sqrt(2))^k for k in Z; here 1 + sqrt(2) is a fundamental unit and (1 + sqrt(2))(sqrt(2) - 1) = 1.
By contrast an imaginary quadratic field has r_1 = 0, r_2 = 1, rank 0 — only finitely many units, the roots of unity.