Basic results in number theory, like the Chinese remainder theorem, the Euclidean algorithm and Euler's theorem, are really about finite structures, namely the rings $\mathbb{Z}/n\mathbb{Z}$ for suitable $n$, or perhaps other finite rings (such as quotients of extensions of $\mathbb{Z}$). Such results can be treated without a worry by a finitist. It is ingredients like this that are sufficient for, say, implementation of the RSA encryption scheme. However, at some point number theoretic results became statements (much more powerful and interesting statements!) about things like rings of adeles.
My question could be taken two ways:
What are the most powerful results in number theory which one can state and prove using the machinery of finite rings?
or
At what point in time did number theory move from considering finite rings to more analytic objects, and by what time was this move 'complete'?
For a reference post, one can point to the Lasker–Noether theorem (1921), which can be taken to be a statement about finitistic objects (and ignoring the possibility that infinite objects exist), namely finite modules $M$ for a ring $R$ (possibly infinite, in which case one could think of it as a finite ring by quotienting by the kernel of $R\to End(M)$).
My motivation for asking about this question is not to be controversial or obtuse, but to get an idea about how far a finitist might get in proving theorems in number theory. Much as reverse mathematics finds the precise strength of a subsystem of second-order arithmetic that is necessary to prove an analytic result (for example, the intermediate value theorem, the Heine–Borel theorem or the Bolzano–Weierstrass theorem), one could try to see how strong theorems in classical number theory are. This of course is something far outside the scope of a single MO question.
Best Answer
Dirichlet's Theorem on Primes in Arithmetic Progressions, proved in 1837, needing real-analytic methods could possibly be the first major candidate for a number-theoretic result departing from finite methods. (This was proved 50 years earlier than Prime Number Theorem).