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Let $q$ be a prime power and $\mathbb{F}_q$ the field with $q$ elements. We know that the places of the global function field $\mathbb{F}_q(x)$ are given by monic, irreducible polynomials in $\mathbb{F}_q[x]$ and $1/x$. In this setting the degree of a place is equal to the degree of the associated polynomial. Hence, we can count the number of places of given degree (Number of monic irreducible polynomials of prime degree $p$ over finite fields).

If we are looking at more general global function fields it is probably more complicated to find the exact number of places with given degree. Thus, my question is:

Are there good estimates of the number of places of a global function field of a given degree?

Severin Schraven
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    What you're asking for is basically a version of the prime number theorem for function fields since places on an algebraic curve will just be prime ideals in the coordinate ring of the curve in an affine patch. You can probably emulate the argument for the prime number theorem over number fields (proved by Landau I think) and just copy it mutatis-mutandis to get a function field version. To spoil you, the split primes will dominate and give you the main term. – Arkady Jul 14 '22 at 03:46
  • @Arkady Thanks! Sounds like a fun exercise for the weekend. – Severin Schraven Jul 14 '22 at 17:52
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    Good luck and please Google “Abstract Analytic Number Theory” if you get stuck. – Arkady Jul 14 '22 at 19:32
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    The number $a_n$ of places of degree $n$ satisfies $a_n=\frac{q^n}{n}+O(\frac{q^{n/2}}{n})$ where $q$ is the cardinality of the constant field, see M. Rosen "Number Theory in Function Fields", Theorem 5.12 – leoli1 Aug 20 '22 at 14:30
  • @leoli1 Thank you for the reference! – Severin Schraven Aug 21 '22 at 06:08

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