TITLE: How does "modern" number theory contribute to further understanding of $\mathbb{N}$?
QUESTION [73 upvotes]: I hope this question is appropriate for MO. It comes from a genuine desire to understand the big picture and ground my own studies "morally".
I'm a graduate student with interest in number theory. I feel like I'm in danger of losing the big picture as I venture a bit deeper and reflect on where I am at. My fundamental is this: I care about the natural numbers - and thus naturally care about the Riemann Zeta function. Number theorists have embarked on various adventures in studying generalized integers (rings of integers of Q-extensions), and their associated zeta functions, and beyond (e.g. Langlands program). Some mathematicians seem to be interested in these generalized integers and zeta functions for their own sake. I am not.
Given my passion for $\mathbb{N}$ and zeta, why should I study these other objects? I understand that philosophically to understand an object it's good to understand its context, and its similarities and differences to its brothers and cousins. This principle makes a lot of sense.
But specifically, what new understandings of $\mathbb{N}$ and zeta have we gained thus far by studying these more general systems? Are there clearly articulated reasons why we can hope to bring back more "treasure" from these more general searches that may shed light on $\mathbb{N}$ in particular? I worry sometimes that number theory is becoming divorced from its original "ground", though I believe (and hope) this feeling derives mainly from ignorance.
EDIT: My question was probably not written very well. I am aware of some of the benefits of studying solutions of polynomials in ring extensions (e.g. solving cases of FLT). My concern is with the broad scope of number theory research today, particularly in the land of generalized zeta-functions and Langlands program. I am uncomfortable (in my ignorance, I admit!) with the apparent lack of a clear connection to the "natural" concerns of number theorists prior to the mid 20th century.
I hope that my question is taken in the spirit of a naive apprentice asking masters for motivation, and a layout of the land of modern research as it connects to concerns that used to be universal.

REPLY [8 votes]: As yet-another comment/question, in addition to the several excellent responses: the question in itself is entirely reasonable, indeed, and is not obviously answered by contemporary professional literature, nor by our current "standard texts". One should rightly suspect/worry that there's been some sort of swindle perpetrated...
And, indeed, the monuments and icons of past times do seem to assume a life and mythology of their own, which all too often has dropped explicit or pointed references to previous history.
So, indeed, one should ask oneself these questions...
And, simultaneously, it would seem fair to ask what all this chatter is about while we've not only not proven RH, but not even proven a zero-free strip!?! Cart before the horse? :)
Probably the genuine explanation in human terms is that the immediate description of things misled us as to what was a cart, and what was a horse, and... maybe, what was "before".
So, yes, while we seem to have made frustratingly little progress on "old issues", and we make only tenuous analogies in further generalizations... what is the alternative?
That is, we do what we can. And we do have some pressure to "hype" it to funding agencies, to Deans, and to each other. Yes, this seems to claim remarkable progress... while we've not really made so much, by more sober standards. :)  But it is not easy to construct a better model of reporting progress.
At some point, then, one observes that it'd be extremely unwise to decide to prove RH for a PhD thesis, or postdoc project, ... or maybe anything else... despite the extreme desirability of progress on exactly such a thing.
One true operational point is that people have to "make a living", which currently requires reporting incredible progress to department heads and deans. Thus, ... noise.
The notion of "deconstruction" of any human activity (that is, acknowledging that it is a human activity, rather than the human-independent, context-independent "absolute" that has a sort of pernicious charm ... for humans) greatly clarifies the seeming paradoxes of human activities, in many cases. :)