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.

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