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.


1 Answer 1


Dirichlet's proof is described in Number Theory: Algebraic Numbers and Functions (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]

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.

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    $\begingroup$ 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 $\mathfrak{o}=\mathbb{Z}\omega_1+\dotsb+\mathbb{Z}\omega_n$. $\endgroup$
    – GH from MO
    Mar 21, 2021 at 16:20
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    $\begingroup$ @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. $\endgroup$ Mar 21, 2021 at 20:17
  • $\begingroup$ @FranzLemmermeyer: Thanks for the clarification. Also, I misspelled Dedekind in the rush (whose name I am very familiar with). $\endgroup$
    – GH from MO
    Mar 22, 2021 at 3:05

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