Irreducible polynomials are often introduced as the analog to prime numbers in polynomial rings. Prime numbers, of course, have a very rich theory, leading to the likes of the Riemann Zeta function and the Prime Number Theorem.


Do any analogs and/or generalizations of primes, such as irreducible polynomials and prime elements, have similarly rich theorems/conjectures?



Primes are to Irreducible Polynomials as Prime-related theorems are to ??
Maybe the first generalization of prime numbers is to prime ideals in algebraic number fields. You do get analogs of the zeta-function, the Prime Number Theorem, even the Riemann Hypothesis. Any text on algebraic number theory will take you there. 


