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?

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    $\begingroup$ Well, there are Frobenius elements built from prime ideals in such a ring of integers. See mathoverflow.net/questions/220598/…. $\endgroup$
    – KConrad
    Feb 4 '18 at 4:49
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    $\begingroup$ I guess much depends on what you call "arithmetic information". For instance, it still makes sense to speak about unramified extensions, and this is ultimately something which depends upon integral structures. $\endgroup$ Feb 4 '18 at 16:59
  • $\begingroup$ @FilippoAlbertoEdoardo I was thinking about things like diaphantine applications e.g. something like Kummers solution of FLT for regular primes, or representability of integers by norm forms etc. but ramification and Frobenius elements are already good points! $\endgroup$ Feb 5 '18 at 14:40

Picking up on the phrase "arithmetic information" in your question, let me give a brief answer coming from logic, although I recognize that this is likely not the answer for which you are looking.

Logicians often measure the arithmetic information content of their mathematical structures by investigating which Turing degrees are encoded within those structures. To my way of thinking, anyone taking the phrase "arithmetic information" seriously will end up talking about the Turing degrees.

For any nonstandard model $\mathcal{M}$ of Peano arithmetic PA, we have the corresponding nonstandard ordered field of rational numbers $\newcommand\Q{\mathbb{Q}}\Q^*$, which has the nonstandard integers $\newcommand\Z{\mathbb{Z}}\Z^*$ sitting inside as a discretely ordered ring, of which it is the quotient field. (But I'm not actually sure whether $\Q^*$ can be algebraic over $\Q$.)

Logicians and researchers in the field known as models of PA measure the arithmetic information content of such a structure by means of the standard system of the model.

The standard system is the set of subsets $A\subseteq\newcommand\N{\mathbb{N}}\N$ that are coded in the model, in the sense that there is an element $n\in \N^*$, usually nonstandard, such that $i\in A$ if and only if the $i^{th}$ prime divides $n$ in $\N^*$. There are a huge variety of coding methods, and all of them are equivalent for the purpose of defining the standard system.

For example, a set $A$ is in the standard system of the model if and only if there is a diophantine equation $p(x,\vec x)=0$ with coefficients in $\Z^*$ such that $i\in A\iff p(i,\vec x)=0$ has a solution in $\Z^*$, for $i\in\N$.

In countable models, a collection $S$ of sets $A\subseteq\N$ is the standard system of a model of PA if and only if $S$ is a Boolean algebra, closed under Turing reduction, and whenever $T\subset 2^{<\N}$ is an infinite binary tree coded in $S$, then there is a branch through $T$ in $S$.

The standard system of a nonstandard model of arithmetic measures the arithmetic information content of the model because it identifies upper bounds on which kind of arithmetic sets could be defined in the model.

For example, some nonstandard models of arithmetic have their standard systems consisting entirely of low sets, which have comparatively little information content; meanwhile, others have a standard system closed under the Turing jump, and so their information content exceeds the halting problem and much more.

A simple compactness argument shows that some nonstandard models of PA have a standard system containing all sets of natural numbers. It remains an open question, however, whether there is a Borel definable such nonstandard model of PA, whose standard system includes all sets $A\subset\N$.


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