Chebotarev density theorem

Relation with Dirichlet's theorem

The Chebotarev density theorem may be viewed as a generalisation of Dirichlet's theorem on arithmetic progressions. A quantitative form of Dirichlet's theorem states that if ${\displaystyle n\ge 2}$ is an integer and ${\displaystyle a}$ is coprime to ${\displaystyle n}$, then the proportion of the primes ${\displaystyle p}$ congruent to ${\displaystyle a}$ mod ${\displaystyle n}$ is asymptotic to ${\displaystyle 1/\varphi (n)}$, where ${\displaystyle \varphi }$ is the Euler totient function.

This is a special case of the Chebotarev density theorem for the ${\displaystyle n}$-th cyclotomic field ${\displaystyle K}$. Indeed, the Galois group of ${\displaystyle K/\mathbb{Q}}$ is abelian and can be canonically identified with the group of invertible residue classes mod ${\displaystyle n}$. The splitting invariant of a prime ${\displaystyle p}$ not dividing ${\displaystyle n}$ is simply its residue class because the number of distinct primes into which ${\displaystyle p}$ splits is ${\displaystyle \varphi (n)/m}$, where ${\displaystyle m}$ is multiplicative order of ${\displaystyle p}$ modulo ${\displaystyle n}$; hence, by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to ${\displaystyle n}$.

Formulation

In their survey article, Lenstra & Stevenhagen (1996) give an earlier result of Frobenius in this area: let ${\displaystyle P(t)}$ be a monic integer polynomial and ${\displaystyle K}$ is a splitting field of ${\displaystyle P(t)}$; then ${\displaystyle K/\mathbb{Q}}$ is a Galois extension. It makes sense to factorise ${\displaystyle P(t)}$ modulo a prime number ${\displaystyle p}$. Its "splitting type" is the list of degrees of irreducible factors of ${\displaystyle P(t)}$ mod ${\displaystyle p}$; i.e. ${\displaystyle P(t)}$ factorizes in some fashion over the prime field ${\displaystyle {\mathbb{F}}_p}$. If ${\displaystyle n}$ is the degree of ${\displaystyle P(t)}$, then the splitting type is a partition ${\displaystyle \mathrm{\Pi }}$ of ${\displaystyle n}$. Now each ${\displaystyle g}$ in ${\displaystyle G=\mathrm{Gal}(K/\mathbb{Q})}$ is a permutation of the roots of ${\displaystyle P(t)}$ in ${\displaystyle K}$; in other words, by choosing an ordering of a root and its algebraic conjugates, ${\displaystyle G}$ is faithfully represented as a subgroup of the symmetric group ${\displaystyle S_n}$. We can write ${\displaystyle g}$ by means of its cycle representation, which gives a "cycle type" ${\displaystyle c(g)}$, again a partition of ${\displaystyle n}$.

The theorem of Frobenius states that for any given choice of ${\displaystyle \mathrm{\Pi }}$ the primes ${\displaystyle p}$ for which the splitting type of ${\displaystyle P(t)}$ mod ${\displaystyle p}$ is ${\displaystyle \mathrm{\Pi }}$ has a natural density ${\displaystyle \delta }$, with ${\displaystyle \delta }$ equal to the proportion of ${\displaystyle g}$ in ${\displaystyle G}$ that have cycle type ${\displaystyle \mathrm{\Pi }}$.

The statement of the more general Chebotarev density theorem is in terms of the Frobenius element of a prime (ideal), which is in fact an associated conjugacy class ${\displaystyle C}$ of elements of the Galois group ${\displaystyle G}$. If we fix ${\displaystyle C}$ then the theorem says that, asymptotically, a proportion ${\displaystyle |C|/|G|}$ of primes have associated Frobenius element as ${\displaystyle C}$. When ${\displaystyle G}$ is abelian, the classes each have size 1. For the case of a non-abelian group of order 6 they have size 1, 2 and 3, and there are correspondingly (for example) ${\displaystyle 50\mathrm{\%}}$ of primes ${\displaystyle p}$ that have an order 2 element as their Frobenius. Thus, these primes have residue degree 2, and so they split into exactly three prime ideals in a degree 6 extension of ${\displaystyle \mathbb{Q}}$ with it as Galois group.

Statement

Let ${\displaystyle L}$ be a finite Galois extension of a number field ${\displaystyle K}$ with Galois group ${\displaystyle G}$. Let ${\displaystyle X}$ be a subset of ${\displaystyle G}$ that is stable under conjugation. The set of primes ${\displaystyle v}$ of ${\displaystyle K}$ that are unramified in ${\displaystyle L}$ and whose associated Frobenius conjugacy class ${\displaystyle F_v}$ is contained in ${\displaystyle X}$ has density ${\displaystyle \mathrm{\#}X/\mathrm{\#}G}$. The statement is valid when the density refers to either the natural density or the analytic density of the set of primes.

Effective version

The generalized Riemann hypothesis implies an effective version of the Chebotarev density theorem: if ${\displaystyle L/K}$ is a finite Galois extension with Galois group ${\displaystyle G}$, and ${\displaystyle C}$ a union of conjugacy classes of ${\displaystyle G}$, the number of unramified primes of ${\displaystyle K}$ of norm below ${\displaystyle x}$ with Frobenius conjugacy class in ${\displaystyle C}$ is

${\displaystyle \frac{|C|}{|G|}(\mathrm{L}\mathrm{i}(x)+O(\sqrt{x}(n\log x+\log |\mathrm{\Delta }|))),}$

where the constant implied in the big-O notation is absolute, ${\displaystyle n}$ is the degree of ${\displaystyle L}$ over ${\displaystyle \mathbb{Q}}$, and ${\displaystyle \mathrm{\Delta }}$ its discriminant.

The effective form of the Chebotarev density theory becomes much weaker without the generalized Riemann hypothesis. Let ${\displaystyle L/K}$ be a finite Galois extension with Galois group ${\displaystyle G}$ and degree ${\displaystyle d}$, take ${\displaystyle \rho }$ to be a nontrivial irreducible representation of ${\displaystyle G}$ of degree ${\displaystyle n}$, and take ${\displaystyle \mathfrak{f}(\rho )}$ to be the Artin conductor of this representation. Suppose that, for ${\displaystyle {\rho }_0}$ a subrepresentation of ${\displaystyle \rho \otimes \rho }$ or ${\displaystyle \rho \otimes \overline{\rho }}$, ${\displaystyle L({\rho }_0,s)}$ is entire; that is, the Artin conjecture is satisfied for all ${\displaystyle {\rho }_0}$. Take ${\displaystyle {\chi }_{\rho }}$ to be the character associated to ${\displaystyle \rho }$. Then there is an absolute positive ${\displaystyle c}$ such that, for ${\displaystyle x\ge 2}$,

${\displaystyle \sum _{p\le x,p\nmid \mathfrak{f}(\rho )}{\chi }_{\rho }({\text{Fr}}_p)\log p=rx+O(\frac{x^{\beta }}{\beta }+x\exp (\frac{-c(dn{)}^{-4}\log x}{3\log \mathfrak{f}(\rho )+\sqrt{\log x}})(dn\log (x\mathfrak{f}(\rho ))),}$

where ${\displaystyle r}$ is ${\displaystyle 1}$ if ${\displaystyle \rho }$ is trivial and is otherwise ${\displaystyle 0}$, and where ${\displaystyle \beta }$ is an exceptional real zero of ${\displaystyle L(\rho ,s)}$; if there is no such zero, the ${\displaystyle x^{\beta }/\beta }$ term can be ignored. The implicit constant of this expression is absolute.

Infinite extensions

The statement of the Chebotarev density theorem can be generalized to the case of an infinite Galois extension ${\displaystyle L/K}$ that is unramified outside a finite set ${\displaystyle S}$ of primes of ${\displaystyle K}$ (i.e. if there is a finite set ${\displaystyle S}$ of primes of ${\displaystyle K}$ such that any prime of ${\displaystyle K}$ not in ${\displaystyle S}$ is unramified in the extension ${\displaystyle L/K}$). In this case, the Galois group ${\displaystyle G}$ of ${\displaystyle L/K}$ is a profinite group equipped with the Krull topology. Since ${\displaystyle G}$ is compact in this topology, there is a unique Haar measure ${\displaystyle \mu }$ on ${\displaystyle G}$. For every prime ${\displaystyle v}$ of ${\displaystyle K}$ not in ${\displaystyle S}$ there is an associated Frobenius conjugacy class ${\displaystyle F_v}$. The Chebotarev density theorem in this situation can be stated as follows:

Let ${\displaystyle X}$ be a subset of ${\displaystyle G}$ that is stable under conjugation and whose boundary has Haar measure zero. Then, the set of primes ${\displaystyle v}$ of ${\displaystyle K}$ not in ${\displaystyle S}$ such that ${\displaystyle F_v\subseteq X}$ has density ${\displaystyle \mu (X)/\mu (G).}$

This reduces to the finite case when ${\displaystyle L/K}$ is finite (the Haar measure is then just the counting measure). A consequence of this version of the theorem is that the Frobenius elements of the unramified primes of ${\displaystyle L}$ are dense in ${\displaystyle G}$.

Important consequences

The Chebotarev density theorem reduces the problem of classifying Galois extensions of a number field to that of describing the splitting of primes in extensions. Specifically, it implies that as a Galois extension ${\displaystyle L/K}$ is uniquely determined by the set of primes of ${\displaystyle K}$ that split completely in it. A related corollary is that if almost all prime ideals of ${\displaystyle K}$ split completely in ${\displaystyle L}$, then in fact ${\displaystyle L=K}$.