Dirichlet's unit theorem as a K1 computation

Remarks

For any commutative ring $A$ the determinant gives a split surjection $K_1(A) \to A^\times$ with kernel $SK_1(A)$, see K1 of a ring and SK1. For $A = \mathcal{O}_F$, Bass, Milnor and Serre proved $SK_1(\mathcal{O}_F) = 0$, so $K_1(\mathcal{O}_F) = \mathcal{O}_F^\times$. The second isomorphism is then Dirichlet's unit theorem.

Example: for $F = \Q(\sqrt{2})$, $K_1(\Z[\sqrt{2}]) \cong \{\pm 1\} \times \Z$, generated by $-1$ and the unit $1 + \sqrt{2}$. For $F = \Q$, $K_1(\Z) = \{\pm 1\}$, the determinants of invertible integer matrices.

Going deeper

For a student

This is the first case of the pattern found by Borel: the rank of $K_n(\mathcal{O}_F)$ depends only on $n$, $r_1$ and $r_2$. For $n = 1$ the logarithmic embedding of units plays the role that Borel's regulator plays for odd $n \ge 3$. Its covolume is the classical regulator that appears in the class number formula.