Chebotarev density theorem

When Carl Friedrich Gauss first introduced the notion of complex integers ${\displaystyle \mathbb {Z} [i]}$, he observed that the ordinary prime numbers may factor further in this new set of integers. He distinguished three cases: if a prime ${\displaystyle p}$ is congruent to 1 mod 4, then it factors into a product of two distinct prime Gaussian integers, or "splits completely"; if ${\displaystyle p}$ is congruent to 3 mod 4, then it remains prime, or is "inert"; and if ${\displaystyle p}$ is 2 then it becomes a product of the square of the prime ⁠${\displaystyle (1+i)}$⁠ and the invertible Gaussian integer ⁠${\displaystyle -i}$⁠; we say that 2 "ramifies". For instance,

${\displaystyle 5=(1+2i)(1-2i)}$ splits completely;

${\displaystyle 3}$ is inert;

${\displaystyle 2=-i(1+i)^{2}}$ ramifies.

From this description, it appears that as one considers larger and larger primes, the frequency of a prime splitting completely approaches ${\displaystyle 50\%}$, and likewise for the primes that remain primes in ${\displaystyle \mathbb {Z} [i]}$. Dirichlet's theorem on arithmetic progressions demonstrates that this is indeed the case. Even though the prime numbers themselves appear rather erratically, splitting of the primes in the extension ${\displaystyle \mathbb {Z} \subset \mathbb {Z} [i]}$ follows a simple statistical law.

Similar statistical laws also hold for splitting of primes in the cyclotomic extensions, obtained from the field of rational numbers by adjoining a primitive root of unity of a given order. For example, the ordinary integer primes group into four classes, each with probability ${\displaystyle 25\%}$, according to their pattern of splitting in the ring of integers corresponding to the 8th roots of unity. In this case, the field extension has degree 4 and is abelian, with the Galois group isomorphic to the Klein four-group. It turned out that the Galois group of the extension plays a key role in the pattern of splitting of primes. Georg Frobenius established the framework for investigating this pattern and proved a special case of the theorem. The general statement was proved by Nikolai Grigoryevich Chebotaryov in 1922.

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 ${\displaystyle n\geq 2}$ 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 ${\displaystyle 1/\phi (n)}$, where ${\displaystyle \phi }$ 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 ${\displaystyle \phi (n)/m}$, 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 ${\displaystyle n}$.

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 ${\displaystyle \Pi }$ of ${\displaystyle n}$. Now each ${\displaystyle g}$ in ${\displaystyle G=\operatorname {Gal} (K/\mathbb {Q} )}$ 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 ${\displaystyle \Pi }$ the primes ${\displaystyle p}$ for which the splitting type of ${\displaystyle P(t)}$ mod ${\displaystyle p}$ is ${\displaystyle \Pi }$ has a natural density ${\displaystyle \delta }$, with ${\displaystyle \delta }$ equal to the proportion of ${\displaystyle g}$ in ${\displaystyle G}$ that have cycle type ${\displaystyle \Pi }$.

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) ${\displaystyle 50\%}$ 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.^[2]

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${\displaystyle \#X/\#G}$.^[3] The statement is valid when the density refers to either the natural density or the analytic density of the set of primes.^[4]

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.^[7] A related corollary is that if almost all prime ideals of ${\displaystyle K}$ split completely in ${\displaystyle L}$, then in fact ${\displaystyle L=K}$.^[8]

• Splitting of prime ideals in Galois extensions
• Grothendieck–Katz p-curvature conjecture