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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.

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    $\begingroup$ You can have a look to this document by Jean-Pierre Serre. $\endgroup$
    – Watson
    Dec 22, 2016 at 18:49
  • $\begingroup$ It's in french. Do you think there is a translation around somewhere. $\endgroup$ Dec 22, 2016 at 20:09

5 Answers 5

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The applications are HUGE! Let me just mention two here.

1) Chebotarev density answers an important question in the study of number fields:

What information is contained within the splitting primes?

To make this slightly more precise, let us define for an extension $L/K$ of number fields the set of splitting primes as follows:

$$\text{Spl}(L/K)=\left\{\mathfrak{p}\text{ a prime of }\mathcal{O}_K:\mathfrak{p}\text{ splits completely in }\mathcal{O}_L\right\}$$

This is an object of extreme importance to us in number theory. In fact, one might say that given an extension $L/K$ our main number theoretic interest in $L$ is to determine $\text{Spl}(L/K)$. So, an obvious question presents itself: what information precisely is contained in the set $\text{Spl}(L/K)$?

The answer is beautiful:

Theorem: Let $L_1,L_2/K$ be be two number field extensions of $K$. Then, the following are equivalent:

  1. $L_1$ and $L_2$ have the same Galois closures.
  2. The sets $\text{Spl}(L_1/K)$ and $\text{Spl}(L_2/K)$ are equal.
  3. The sets $\text{Spl}(L_1/K)$ and $\text{Spl}(L_2/K)$ are almost equal.

Here 'almost equal' means that there are only finitely many primes not contained in either.

So, this theorem is INCREDIBLE! It tells you that the number theoretic question we have always been interested, which primes split completely, is not just a question of number theoretic significance, but of field theoretic significance. In particular, to know a field's Galois closure (field theory) is the same thing as knowing its set of split primes (number theory).

This also shows that while, a priori, knowing just the split primes only tells you some number theoretic information it doesn't tell you all. Namely, the splits primes shouldn't, a priori, tell you about ramified primes, etc. But, if your extension is Galois, then the above tells you that the split primes know about $L$ itself, and so, of course, know about the other number theoretic data.

OK, excellent, this is a beautiful theorem. What does it have to do with Chebotarev density? Well—everything! Namely, the proof of this theorem is essentially Chebotarev density. Let me give a sketch below:

Proof: Suppose first that $L_1$ and $L_2$ have the same Galois closure, call it $L$. Then, elementary algebraic number theory shows that

$$\text{Spl}(L_1/K)=\text{Spl}(L/K)=\text{Spl}(L_2/K)$$

which shows that 1. implies 2.

Conversely, suppose that $\text{Spl}(L_1/K)=\text{Spl}(L_2/K)$. Then, if $L_i'$ denotes the Galois closures of $L_i$, then $\text{Spl}(L_1'/K)=\text{Spl}(L_2'/K)$.

But, note then that, again by basic number theory, this implies that

$$\text{Spl}(L_1'/K)=\text{Spl}(L_1'L_2'/K)=\text{Spl}(L_2'/K)$$

But, by considering Chebotarev density, since all of these extensions are Galois, we deduce that the following densities are equal

$$\frac{1}{[L_1':K]}=\frac{1}{[L_1'L_2':K]}=\frac{1}{[L_2':K]}$$

which, in particular, shows that $[L_i':K]=[L_1'L_2':K]$ which implies that $L_1'=L_1'L_2'=L_2'$ which shows that $L_1'=L_2'$ as desired. $\blacksquare$

As an application of this theorem, we deduce the main idea of Class Field Theory:

Idea: Extensions of $K$ for which data about the extension is entirely 'internal to K' are precisely the abelian extensions.

A rigorous example of this is:

Theorem: If $K/\mathbb{Q}$ is a Galois extension such that the splitting behavior is determined $\mod N$ for some integer $N$, then $K$ is abelian (in fact contained in $\mathbb{Q}(\zeta_N)$).

If you want to read more about this application, you can see my blog post here.

2) The second idea comes from the theory of Galois representations. Namely, let us say that an $\ell$-adic Galois representation of $K$ (a number field) is a continuous group homomorphism

$$\rho:G_K\to\text{GL}_n(\overline{\mathbb{Q}_\ell})$$

These are EXTREMELY important in modern number theory, in ways in which I won't go into here. But, an interesting question is how little information is needed to determine $\rho$. What data do we need to compute to know that we've uniquely characterized $\rho$?

If $\rho$ is unramified almost everywhere, the answer is very satisfying. We say that $\rho$ is unramified almost everywhere if it factors through $G_{K,S}=\text{Gal}(K^S/K)$, where $K^S$ is the maximal extension of $K$ unramified outside of $S$, for some finite set $S$ of primes of $K$.

These are the most important types of Galois representations, one generally only considers representations of this form. In particular all 'geometric Galois representations', those coming from geometry (e.g. the Tate module of an abelian variety), are unramified almost everywhere.

The result is then the following:

Theorem: Let $\rho_1:G_{K,S}\to\text{GL}_n(\overline{\mathbb{Q}_\ell})$ and $\rho_2:G_{K,T}\to\text{GL}_n(\overline{\mathbb{Q}_\ell})$be two unramified almost everywhere Galois representations. Then, $\rho_1=\rho_2$ if and only if

$$\text{tr}(\rho_1(\text{Frob}_\mathfrak{p}))=\text{tr}(\rho_2(\text{Frob}_\mathfrak{p}))$$

for all $\mathfrak{p}\notin S\cup T$.

Let me just explain the above notation. For $\mathfrak{p}\notin S$ there is a well-defined Frobenius conjugacy class $\text{Frob}_\mathfrak{p}\in G_{K,S}$. Indeed, one can understand this in elementary terms as writing $K^S$ as a union of finite extensions of $K$. Then, since each of these extensions are unramified at $\mathfrak{p}$ they have a Frobenius conjugacy class, and so we obtain one in the union.

Then, $\text{tr}(\rho(\text{Frob}_\mathfrak{p}))$ denotes the trace of the image of any element of $\text{Frob}_\mathfrak{p}\subseteq G_{K,S}$. It, of course, is independent of choice since the trace function ignores conjugation.

Thus, this theorem tells us that the huge amount of data encompassed in $\rho$ is, in fact, contained in this MUCH smaller set of the traces of the Frobenii. Amazing!

The proof relies on two facts:

a) The Brauer-Nesbitt theorem.

b) The fact that the Frobenius conjugacy classes $\{\text{Frob}_\mathfrak{p}\}_{\mathfrak{p}\notin S}$ are dense in $G_{K,S}$.

The first of these is just a classic result in algebra. But, b) is, for all intents and purposes the 'same thing' as Chebotarev density. Exercise: use Chebotarev density to prove b)!

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    $\begingroup$ Exercise: add the supplementary hypothesis to the "theorem" to make it true. $\endgroup$
    – Infinity
    Mar 30, 2018 at 3:32
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    $\begingroup$ @Infinity You mean because I forgot semisimplification? You can edit if you'd like. I'm on mobile. $\endgroup$ Apr 10, 2018 at 20:13
  • $\begingroup$ Is it possible to prove the results related to CFT (like the one in your blog or the first application you mention) without using CDT? One possible hurdle could be that CFT only talks about abelian extensions. $\endgroup$ May 27, 2021 at 5:11
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    $\begingroup$ @mathemather you don't need CDT for the first application above. Using Dirichlet density instead of natural density, Kronecker showed for Galois $L/K$ that the Dirichlet density of primes in ${\rm Spl}(L/K)$ is $1/[L:K]$ in the 19th century (CDT was first proved in the 1920s), and that suffices to prove the equivalence of the three conditions in the first application. A prime splits completely in a Galois extension exactly when its Frobenius conjugacy class is trivial, and working out the density of primes with a trivial Frobenius conjugacy class does not require the full CDT. $\endgroup$
    – KCd
    Apr 16, 2023 at 0:49
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    $\begingroup$ @mathemather Kronecker's work on this topic (in 1880) led him to conjecture a result about Dirichlet densities of certain prime numbers that was proved by Frobenius that was later refined to the CDT, first for finite Galois extensions of $\mathbf Q$ and later for finite Galois extensions of any number field. $\endgroup$
    – KCd
    Apr 16, 2023 at 0:54
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One of my favorite applications is a simple one to class groups and unique factorization in rings of integers.

Fix a number field $K/\mathbf{Q}$. In a first course on algebraic number theory, you encounter the class group $\mathrm{Cl}(K)$ and prove that $\mathcal{O}_K$ has unique factorization if and only if $h_K = \# \mathrm{Cl}(K) = 1$. This is often followed by a statement akin to "the class group measures the failure of unique factorization to hold." But the result above only says something about whether $h_k = 1$ or $h_K > 1$; the informal statement implies that somehow the bigger $h_K$ is, unique factorization should somehow fail "more often".

This can be made precise with class field theory and Chebotarev. Let $H/K$ be the Hilbert class field of $K$, so class field theory gives a canonical isomorphism $\mathrm{Cl}(K) \overset{\sim}{\longrightarrow} \mathrm{Gal}(H/K)$ that takes a prime $\mathfrak{p}$ to its corresponding Frobenius element. In particular, a prime is totally split in $H/K$ if and only if it is principal. But Chebotarev says that the totally split primes have density $\frac{1}{\# \mathrm{Gal}(H/K)} = \frac{1}{\# \mathrm{Cl}(K)} = \frac{1}{h_K}$, so the density of principal primes is the inverse of the class number!

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    $\begingroup$ This is a super nice point! Thanks for making it. $\endgroup$ Feb 28, 2016 at 0:34
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A very important application of CDT in number theory concerns Artin's conjecture on primitive roots, namely that for every non-square integer $a\neq0,\pm1$ there exist infinitely many primes $p$ for which $\mathbb{F}_p^*=\langle a\rangle$, i.e. $a$ is a primitive root modulo $p$. This conjecture has been proved under GRH by Hooley and the main tool used was in fact CDT (together with its error term which under GRH becomes suitable for the sums involved).

The main concept: $a$ is NOT a primitive root mod $p$ if for some $q\mid(p-1)$ you have $a^{(p-1)/q}\equiv 1$ mod $p$ and this is equivalent in saying that the prime $p$ splits completely in the Kummerian extension $\mathbb{Q}(\zeta_q,a^{1/q})/\mathbb{Q}$. Then, if you want $a$ to be a primitive root mod $p$ you have, by CDT, a probability $$ 1-\frac1{[\mathbb{Q}(\zeta_q,a^{1/q}):\mathbb{Q}]}\;. $$ For Artin's conjecture, you think of fixing a certain $q$ and look after those primes $p$ which don't split completely in the related Kummer extension. Now, keep in mind that this is just the basic idea behind Hooley's proof, since you should be aware of using the inclusion-exclusion principle to avoid multiple counting for those primes which don't split in some extensions, the fact that the events "$p$ does not split completely in $K_1/\mathbb{Q}$'' and "$p$ does not split completely in $K_2/\mathbb{Q}$'' are not in general independent and, most important, when $q$ tends to infinity you have an infinite sum whose behaviour depends on the error term of the CDT: if you assume GRH you can handle the errors, otherwise they overwhelm the main term you're interested in.

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I will try to give an elementary application I came across. Suppose $f(x)\in \mathbb{Z}[x]$ is an irreducible polynomial of degree at least $2$. We will show that there are infinitely many primes $p\in \mathbb{Z}$ such that $\overline{f}\in \mathbb{Z}/p\mathbb{Z}[x]$ has no root.

Let $K=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$ where $\alpha_i$ are distinct roots of $f(x)$ in $\mathbb{C}$. The Galois group $Gal(K/\mathbb{Q})$ acts transitively on roots $\lbrace \alpha_i\rbrace$ and hence defines a unique permutation of $n$ elements. Hence we may think of $Gal(K/\mathbb{Q})$ as a subgroup of $S_n$. For $T_1=\mathbb{Q}(\alpha_1)$ we see that $Gal(K/T_1)$ is a proper subgroup of $Gal(K/\mathbb{Q})$ and its conjugates $\sigma Gal(K/T_1)\sigma^{-1}=Gal(K/T_i)$ where $\sigma(\alpha_1)=\alpha_i$. As a finite group cannot be written as union of conjugates of a proper subgroup, find $\sigma \notin \bigcup Gal(K/T_i)$. By CDT, there are infinitely many primes $p\in \mathbb{Z}$ such that there exists a prime $\mathfrak{p}$ above $p$ and $\left[\frac{K/\mathbb{Q}}{\mathfrak{p}}\right]=\sigma$. The cycle type of $\sigma$ tells us the irreducible decomposition of $\overline{f}$ in the quotient field. For $p$ as above, if there was a root of $\overline{f}$ in the residue field of $p$, then $\sigma$ must fix a root of $f$ i.e for some $i$ we must have $\sigma(\alpha_i)=\alpha_i$, which implies $\sigma \in Gal(K/T_i)$. This contradicts our choice of $\sigma$.

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    $\begingroup$ Nice answer! Chebotarev also shows, categorically, that in any extension there are infinitely many split primes. Using this, one can see that any irreducible polynomial $f\in\mathbb{Z}[x]$ must have $\deg(f)$ distinct roots mod infinitely many $p$. $\endgroup$ Feb 27, 2016 at 13:54
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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.)

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