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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 '11 at 22:38
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closed as not a real question by Gjergji Zaimi, GH from MO, Zev Chonoles, Andres Caicedo, Martin Brandenburg Oct 13 '11 at 7:46

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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.

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From there on to function fields it's but a jump. –  Felipe Voloch Oct 13 '11 at 12:53
    
A step over a curb type jump or an Evel Knievel Snake River Canyon-like jump? Gerhard "Inquiring Minds Want To Gauge" Paseman, 2011.10.13 –  Gerhard Paseman Oct 13 '11 at 20:33
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