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## Primes are to Irreducible Polynomials as Prime-related theorems are to ?? [closed]

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?

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Well, there is a zeta function and a prime number theorem for irreducible polynomials over finite fields, but both are quite easy to investigate. – Qiaochu Yuan Oct 12 2011 at 22:38