The fact that the ring of integers in a finite extension of $\Bbb Q$ is a Dedekind domain and purely algebraic properties of Dedekind domains are absolutely essential for algebraic number theory. So it seems natural to ask about infinite extensions next.
While the (not necessarily absolute) Galois groups of infinite algebraic extensions play a role e.g. in Iwasawa theory, it seems to me that the rings of integers of infinite algebraic extensions receive little attention. (Though this may just be a lack of knowledge on my part.)
There is of course the classical result due to Dedekind that the ring of all algebraic integers is a Bezout domain. There are also notions in commutative algebra to describe what kind of ring the integral closure in an infinite algebraic extension will be. (The fact that one may call Prüfer domains "arithmetical domains" makes this question seem more pertinent.)
But while knowing for example the class group of the ring of integers in a finite extension can have striking consequences, it is not obvious to me what kind of arithmetic information we get for infinite algebraic extensions when we know for example class group of the ring of integers or that it is a Bezout domain. And I've never seen general results about Prüfer or Bezout domains applied in algebraic number theory.
So my question is the question in the title: What kind of arithmetic information can we get out of algebraic structure of the ring of integers in an algebraic extension? Do you know examples where results about Prüfer domains/Bezout domains/Krull domains etc. are applied to number theory?