# Chebotarev's density theorem

**Chebotarev's density theorem** in algebraic number theory describes statistically the splitting of primes in a given Galois extension *K* of the field *Q* of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of *K*. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime *p* in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes *p* less than a large integer *N*, tends to a certain limit as *N* goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in Template:Harv.

A special case that is easier to state says that if *K* is an algebraic number field which is a Galois extension of *Q* of degree *n*, then the prime numbers that completely split in *K* have density

- 1/
*n*

among all primes. More generally, splitting behavior can be specified by assigning to (almost) every prime number an invariant, its Frobenius element, which strictly is a representative of a well-defined conjugacy class in the Galois group

*Gal*(*K*/*Q*).

Then the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conjugacy class with *k* elements occurs with frequency asymptotic to

*k*/*n*.

## History and motivation

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

From this description, it appears that as one considers larger and larger primes, the frequency of a prime splitting completely approaches 1/2, and likewise for the primes that remain primes in *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

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 1/4, 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.

## 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 *N*≥*2* is an integer and *a* is coprime to *N*, then the proportion of the primes *p* congruent to *a* mod *N* is asymptotic to 1/*n*, where *n*=φ(*N*) is the Euler totient function. This is a special case of the Chebotarev density theorem for the *N*th cyclotomic field *K*. Indeed, the Galois group of *K*/*Q* is abelian and can be canonically identified with the group of invertible residue classes mod *N*. The splitting invariant of a prime *p* not dividing *N* is simply its residue class because the number of distinct primes into which *p* splits is φ(*N*)/m, where m is multiplicative order of *p* modulo *N;* hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to *N*.

## Formulation

Template:Harvtxt give an earlier result of Frobenius in this area. Suppose *K* is a Galois extension of the rational number field **Q**, and *P*(*t*) a monic integer polynomial such that *K* is a splitting field of *P*. It makes sense to factorise *P* modulo a prime number *p*. Its 'splitting type' is the list of degrees of irreducible factors of *P* mod *p*, i.e. *P* factorizes in some fashion over the prime field **F**_{p}. If *n* is the degree of *P*, then the splitting type is a partition Π of *n*. Considering also the Galois group *G* of *K* over **Q**, each *g* in *G* is a permutation of the roots of *P* in *K*; in other words by choosing an ordering of α and its algebraic conjugates, *G* is faithfully represented as a subgroup of the symmetric group *S*_{n}. We can write *g* by means of its cycle representation, which gives a 'cycle type' *c*(*g*), again a partition of *n*.

The *theorem of Frobenius* states that for any given choice of Π the primes *p* for which the splitting type of *P* mod *p* is Π has a natural density δ, with δ equal to the proportion of *g* in *G* that have cycle type Π.

The statement of the more general *Chebotarev theorem* is in terms of the Frobenius element of a prime (ideal), which is in fact an associated conjugacy class *C* of elements of the Galois group *G*. If we fix *C* then the theorem says that asymptotically a proportion |*C*|/|*G*| of primes have associated Frobenius element as *C*. When *G* is abelian the classes of course 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) 50% of primes *p* that have an order 2 element as their Frobenius. So these primes have residue degree 2, so they split into exactly three prime ideals in a degree 6 extension of *Q* with it as Galois group.^{[1]}

## Statement

Let *L* be a finite Galois extension of a number field *K* with Galois group *G*. Let *X* be a subset of *G* that is stable under conjugation. The set of primes *v* of *K* that are unramified in *L* and whose associated Frobenius conjugacy class *F*_{v} is contained in *X* has density

### Infinite extensions

The statement of the Chebotarev density theorem can be generalized to the case of an infinite Galois extension *L* / *K* that is unramified outside a finite set *S* of primes of *K* (i.e. if there is a finite set *S* of primes of *K* such that any prime of *K* not in *S* is unramified in the extension *L* / *K*). In this case, the Galois group *G* of *L* / *K* is a profinite group equipped with the Krull topology. Since *G* is compact in this topology, there is a unique Haar measure μ on *G*. For every prime *v* of *K* not in *S* there is an associated Frobenius conjugacy class *F*_{v}. The Chebotarev density theorem in this situation can be stated as follows:^{[2]}

- Let
*X*be a subset of*G*that is stable under conjugation and whose boundary has Haar measure zero. Then, the set of primes*v*of*K*not in*S*such that*F*_{v}⊆ X has density

This reduces to the finite case when *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 *L* are dense in *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 of *K*, *L* is uniquely determined by the set of primes of *K* that split completely in it.^{[3]} A related corollary is that if almost all prime ideals of *K* split completely in *L*, then in fact *L* = *K*.^{[4]}

## Notes

## References

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