# How Dirichlet proved Dirichlet's unit theorem for general number fields?

For a general number field $$K$$, Dirichlet's unit theorem states that the unit group of the ring of integers of $$K$$ is a finitely generated group of rank $$r_1+r_2-1$$.

It seems that standard algebraic number theory textbooks usually prove this theorem by using Minkowski's theory or using the Blichfeldt-Minkowski Lemma to show that the group of norm 1 ideles modulo $$K^\times$$ is compact. Since both Minkowski and Blichfeldt were born significantly later than Dirichlet, I wonder that Dirichlet's proof may be different from the above two?

I should say that, for real quadratic fields, or equivalently, Pell's equation $$x^2-dy^2=1$$, I have saw an argument in Dirichlet's book "Lectures on number theory" where the author first shows that, for the irrational number $$\sqrt{d}$$ there are infinitely many rational numbers $$\frac{p}{q}$$ such that $$|\frac{p}{q} -\sqrt{d}| \leq \frac{1}{q^2}.$$ Using this result, one can easily produce infinitely many principal ideals with bounded norms of the ring of integers $$\mathbb{Q}(\sqrt{d})$$ whence produce a nontrivial unit. Done!

To me this argument is distinct from the Minkowski's method in some sense and I am unable to generalize the above argument to prove the Dirichlet's unit theorem for general number fields. So I wonder that how Dirichlet proved his theorem for general number fields. Thanks.

Dirichlet did not use Minkowski’s theorem; he proved the unit theorem in 1846 while Minkowski’s theorem appeared in 1889. Dirichlet’s substitute for the convex-body theorem was the pigeonhole principle. Dirichlet did not state the unit theorem for all orders, but only those of the form $$\mathbf{Z}[\alpha]$$, since at the time these were the kinds of rings that were considered. [source]
• Last night I looked into Dirichlet's Vorlesungen über Zahlentheorie, with Dedkind's supplement. There the full ring of integers is considered, given in the form of $\mathfrak{o}=\mathbb{Z}\omega_1+\dotsb+\mathbb{Z}\omega_n$. Mar 21, 2021 at 16:20
• @GH: This is of course due to Dedekind, who rewrote Dirichlet's proofs for the maximal order. As a matter of fact, the unit theorem for orders ${\mathbb Z}[\alpha]$ implies and follows from the unit theorem for the maximal order. Mar 21, 2021 at 20:17