TITLE: What are some applications of Chebotarev Density Theorem?
QUESTION [22 upvotes]: Let $L/K$ be a Galois extension of number fields and let $\mathcal{C}$ be a conjugacy class in $Gal(L/K)$. Let $\mathbb{P}(K)$ be the set of all prime ideals in $K$ and let $\left(\frac{L/K}{\mathfrak{p}} \right)$ correspond to the associated conjugacy class of Frobenius elements living over $\mathfrak{p}$(of course unramified) and suppose $A=\left\lbrace \mathfrak{p}\in P(K) \mid \left(\frac{L/K}{\mathfrak{p}} \right)=\mathcal{C} \right\rbrace$.
Then the Chebotarev Density Theorem states that $\delta(A)=\frac{|C|}{[L:K]}$. 
This also is a generalisation of Frobenius density theorem. 
For positive integers $a,n$ such that $\gcd(a,n)=1$ CDT for $K=\mathbb{Q}$ and $L=\mathbb{Q}(\zeta_n)$ and $\mathcal{C}=\lbrace \zeta_n \to \zeta_n^a \rbrace$ gives Dirichlet's theorem of infinitude of primes in arithmetic progression. 
I wish to ask what other applications are of this theorem.

REPLY [3 votes]: In 1996 Pascal Koiran showed that given a system $F=0$ of $k$ polynomial equations on $n$ variables, where the maximal degree over all monomials is $D$ and the bit-size of the largest coeffient is $h$, one can determine if the system has a solution $F=0$ over $\mathbb{C}^n$, by showing that it has a solution in $\mathbb{Z}/p \mathbb{Z}$ for many primes $p$.
In more detail, if the system does not have a solution in $\mathbb{C}^n$, then unconditionally, by Effective Nullstellensatz, there are at most
$$A_F=4n(n+1)D^n(h+\log k + (n+7)\log(n+1)D$$
primes $p$ such that $F=0\mod p$.
Likewise, if the system does have a solution in $\mathbb{C}^n$, then by the prime ideal theorem, unconditionally there is a positive density of primes $p$ modulo which there is a solution.
The denouement is that, conditioned on the Generalized Riemann Hypothesis, by Effective Chebotarev Density, Koiran showed that these primes $p$ are distributed "evenly" enough such that one can apply standard tricks of universal hashing to give a small certificate that the system $F=0$ is likely to be satisfiable modulo more than $2A_F$ primes $p$, and thus is likely to be satisfiable in $\mathbb{C}^n$.  Essentially, one finds a prime $q$ such that $F$ is satisfiable in $\mathbb{Z}/q \mathbb{Z}$ and $H(q)=0$ for some nice random hash function $H$, thus showing there are likely enough primes $q$ to invert $H$, thus likely enough primes modulo which $F=0$ has a solution.
As a bonus that's an answer in its own right, in 2011 Kuperberg applied Koiran's results to give a small certificate of knottedness for a knot diagram (again conditioned on Effective Chebotarev by way of the GRH.)