For any course in algebraic number theory, one must prove the finiteness of class number and also Dirichlet's unit theorem. The standard proof uses Minkowski's theorem. Is there a way to avoid it?
The reasons I am asking this question are the following.
$1$. Minkowski lived long after Dirichlet and Dedekind(esp Dirichlet). So the original proof cannot likely have used Minkowski's theorem as such. If the original proof did use Minkowski's theorem, then it was of course found by someone else, most probably Dirichlet, and it is unfair to use the name Minkowski's theorem.
$2$. Even more importantly, the finiteness of classnumber and some version of unit theorem is true(at least I hope so) for all global fields. And there of course one cannot talk of Minkowski's theorem.
The objection I have for Minkowski's theorem is that it seems to be ad hoc, coming out of nowhere. And it seems that not much work is going on nowadays in the subject of geometry of numbers.
So it will be really nice to have a method which would feel more natural and is perhaps more general.