I recently learned the Chebotarev Density Theorem for global fields.

As far as I have seen, all applications of CDT seem to focused around proving some set of prime ideals (or places in function field case) is infinite.

I wished to ask whether there is any quantitative aspect to this. While the existence of density $\delta(A)$ and being non-zero is enough to say that $A$ is infinite, what purpose does the number $\delta(A)$ serve.

More precisely, would someone care to give an application where we somehow use the density.

  • $\begingroup$ Could you please include the exact result you care about. Some might not know it, and some others might know several versions. $\endgroup$ – quid Jan 18 '16 at 14:45
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    $\begingroup$ As an example, the proof of Dirichlet's theorem of primes in arithmetic progressions using Chebotarev tells us not only that the set of primes $\{p : p\equiv a \pmod N\},\, (a,N) = 1$ is infinite, but also that the primes modulo $N$ are evenly distributed among the $\phi(N)$ classes. $\endgroup$ – Mathmo123 Jan 18 '16 at 14:47
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    $\begingroup$ The CDT has a main role in the study of Artin's conjecture for primitive roots, since it's a fundamental tool which allows to relate the density of primes $p$ for which a certain non-square integer $a\neq 0,\pm 1$ is a primitive root modulo $p$ to the density of primes splitting completely in the Kummer extensions $\mathbb{Q}(a^{1/k},\zeta_k)$: to express the latter density, we need CDT. Artin's conjecture has been proved under GRH, since in this case the error in the CDT is small enough to not spoil the asymptotic estimates, see e.g. the memorable paper by Hooley "On Artin's conjecture". $\endgroup$ – PITTALUGA Jan 19 '16 at 11:59
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    $\begingroup$ The density of primes in $K$ that are principal is $\frac{1}{h_K}$. This follows from class field theory and Chebotarev. $\endgroup$ – Brandon Carter Jan 20 '16 at 3:37
  • $\begingroup$ One can generalize the above comment: the set of prime ideals of $O_K$ in a given ideal class of the class group $C_K$ is a set of positive natural density $1/|C_K|$. $\endgroup$ – Watson Dec 2 '18 at 10:15

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