I thought of this question while I was day dreaming about twin primes, and is a very naive, perhaps stupid question.
Since we can generalize the notion of a prime number to prime ideals of rings, and arithmetic geometry revolves around tackilng number theory problems by looking at their "higher dimensional analogues" I starting thinking whether or not we could order prime ideals in number fields and extract some results in the distribution of primes but by carrying our analysis to general number fields.
I had a Set Theory class long ago and my knowledge on that is very rusty.I don't even know if we can order number fields in a way such that order relation resembles the one in the integers, or even if we could, would that order relation be compatible with the field structure?
In the little I have studied so far about the subject, algebraic number theory doesn't seem to deal with problems in the distribution of primes, as far as I know only analytic number theory does that although at an advanced level it relies heavily on algebraic geometry.
Is there any part of algebraic number theory/ arithmetic geoemtry that focus on problems such like the distribution of primes?