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.



How Dirichlet proved Dirichlet's unit theorem for general number fields?
Dirichlet's proof is described in [Number Theory: Algebraic Numbers and Functions](https://books.google.nl/books?id=qEwpwWyVPIAC) (starting on page 48).


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]](https://kconrad.math.uconn.edu/blurbs/gradnumthy/unittheorem.pdf)


There is an oft-repeated story that the idea for the proof came to Dirichlet while he was attending Easter mass in the Sistine Chapel in Rome. Attempts to document that story are described [in this MO question](https://mathoverflow.net/q/106196/11260).


