{{short description|Describes statistically the splitting of primes in a given Galois extension of Q}} In mathematics, specifically in algebraic number theory, the '''Chebotarev density theorem''', named after Nikolai Chebotarev, statistically describes the splitting of primes in a given Galois extension <math>K</math> of the field <math>\mathbb{Q}</math> of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of <math>K</math>. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime <math>p</math> 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 <math>p</math> less than a large integer <math>N</math>, tends to a certain limit as <math>N</math> goes to infinity. It was proved by Chebotarev in his thesis in 1922.<ref>{{citation |first=Nikolai |last=Tschebotareff |journal=Mathematische Annalen |volume=95 |year=1926 |pages=191–228 |doi= 10.1007/BF01206606 |title=Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören|trans-title=Determining the density of a set of prime numbers belonging to a given substitution class|language=de}}</ref>
A special case that is easier to state says that if <math>K</math> is an algebraic number field which is a Galois extension of <math>\mathbb{Q}</math> of degree <math>n</math>, then the prime numbers that completely split in <math>K</math> have density <math>1/n</math> among all primes. More generally, splitting behavior can be specified by assigning to (almost) every prime number an invariant, its Frobenius element, which is a representative of a well-defined conjugacy class in the Galois group <math>\operatorname{Gal}(K/\Q)</math>. In this case, the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conjugacy class with <math>k</math> elements occurs with frequency asymptotic to <math>k/n</math>.
== History and motivation ==
When Carl Friedrich Gauss first introduced the notion of complex integers <math>\Z[i] </math>, he observed that the ordinary prime numbers may factor further in this new set of integers. He distinguished three cases: if a prime <math>p</math> is congruent to 1 mod 4, then it factors into a product of two distinct prime Gaussian integers, or "splits completely"; if <math>p</math> is congruent to 3 mod 4, then it remains prime, or is "inert"; and if <math>p</math> is 2 then it becomes a product of the square of the prime {{tmath|(1+i)}} and the invertible Gaussian integer {{tmath|-i}}; we say that 2 "ramifies". For instance,
: <math> 5 = (1 + 2i)(1-2i) </math> splits completely; : <math> 3 </math> is inert; : <math> 2 = -i(1+i)^2 </math> ramifies.
From this description, it appears that as one considers larger and larger primes, the frequency of a prime splitting completely approaches <math>50% </math>, and likewise for the primes that remain primes in <math>\Z[i] </math>. 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 <math> \mathbb{Z}\subset \mathbb{Z}[i] </math> 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 <math>25% </math>, 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 ''<math>n \geq 2 </math>'' is an integer and <math>a </math> is coprime to ''<math>n </math>'', then the proportion of the primes <math>p </math> congruent to <math>a </math> mod ''<math>n </math>'' is asymptotic to ''<math>1/\phi(n) </math>'', where ''<math>\phi </math>'' is the Euler totient function.
This is a special case of the Chebotarev density theorem for the ''<math>n </math>''-th cyclotomic field ''<math>K </math>''. Indeed, the Galois group of ''<math>K/\Q </math>'' is abelian and can be canonically identified with the group of invertible residue classes mod ''<math>n </math>''. The splitting invariant of a prime ''<math>p </math>'' not dividing ''<math>n </math>'' is simply its residue class because the number of distinct primes into which <math>p </math> splits is ''<math>\phi(n)/m </math>'', where ''<math>m </math>'' is multiplicative order of <math>p </math> modulo ''<math>n </math>;'' hence, by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to ''<math>n </math>''.
==Formulation==
In their survey article, {{harvtxt|Lenstra|Stevenhagen|1996}} give an earlier result of Frobenius in this area: let <math>P(t) </math> be a monic integer polynomial and <math>K </math> is a splitting field of <math>P(t) </math>; then <math>K/\Q </math> is a Galois extension. It makes sense to factorise <math>P(t) </math> modulo a prime number <math>p </math>. Its "splitting type" is the list of degrees of irreducible factors of <math>P(t) </math> mod <math>p </math>; i.e. <math>P(t) </math> factorizes in some fashion over the prime field <math>\mathbb{F}_p </math>. If ''<math>n </math>'' is the degree of <math>P(t) </math>, then the splitting type is a partition ''<math>\Pi </math>'' of ''<math>n </math>''. Now each <math>g</math> in <math>G = \operatorname{Gal}(K/\Q)</math> is a permutation of the roots of <math>P(t) </math> in <math>K </math>; in other words, by choosing an ordering of a root and its algebraic conjugates, <math>G</math> is faithfully represented as a subgroup of the symmetric group <math>S_n </math>. We can write <math>g</math> by means of its cycle representation, which gives a "cycle type" ''<math>c(g) </math>'', again a partition of ''<math>n </math>''.
The theorem of Frobenius states that for any given choice of ''<math>\Pi </math>'' the primes <math>p </math> for which the splitting type of <math>P(t) </math> mod <math>p </math> is ''<math>\Pi </math>'' has a natural density <math>\delta </math>, with <math>\delta </math> equal to the proportion of <math>g</math> in <math>G </math> that have cycle type ''<math>\Pi </math>''.
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 <math>C </math> of elements of the Galois group <math>G</math>. If we fix <math>C </math> then the theorem says that, asymptotically, a proportion <math>|C|/|G|</math> of primes have associated Frobenius element as <math>C </math>. When <math>G</math> 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) <math>50% </math> of primes <math>p </math> 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 <math>\Q </math> with it as Galois group.<ref>This particular example already follows from the Frobenius result, because ''G'' is a symmetric group. In general, conjugacy in ''G'' is more demanding than having the same cycle type.</ref>
==Statement== Let <math>L </math> be a finite Galois extension of a number field <math>K </math> with Galois group <math>G </math>. Let <math>X </math> be a subset of <math>G </math> that is stable under conjugation. The set of primes <math>v </math> of <math>K </math> that are unramified in <math>L </math> and whose associated Frobenius conjugacy class <math>F_v </math> is contained in <math>X </math> has density<math>\#X/\#G</math>.<ref name="Section">Section I.2.2 of Serre</ref> The statement is valid when the density refers to either the natural density or the analytic density of the set of primes.<ref>{{cite web |url= http://websites.math.leidenuniv.nl/algebra/Lenstra-Chebotarev.pdf|title=The Chebotarev Density Theorem |last=Lenstra |first=Hendrik |date=2006 |access-date=7 June 2018 }}</ref>
===Effective version=== The generalized Riemann hypothesis implies an effective version<ref>{{cite journal|first1=J.C.|last1=Lagarias|first2=A.M.|last2=Odlyzko|title=Effective Versions of the Chebotarev Theorem|journal=Algebraic Number Fields|year=1977|pages=409–464}}</ref> of the Chebotarev density theorem: if <math>L/K </math> is a finite Galois extension with Galois group <math>G </math>, and <math>C </math> a union of conjugacy classes of <math>G </math>, the number of unramified primes of <math>K </math> of norm below <math>x </math> with Frobenius conjugacy class in <math>C </math> is :<math>\frac{|C|}{|G|}\Bigl(\mathrm{Li}(x)+O\bigl(\sqrt x(n\log x+\log|\Delta|)\bigr)\Bigr),</math> where the constant implied in the big-O notation is absolute, <math>n </math> is the degree of <math>L </math> over <math>\Q </math>, and <math>\Delta </math> its discriminant.
The effective form of the Chebotarev density theory becomes much weaker without the generalized Riemann hypothesis. Let <math>L/K </math> be a finite Galois extension with Galois group <math>G </math> and degree <math>d</math>, take <math>\rho</math> to be a nontrivial irreducible representation of <math>G </math> of degree <math>n </math>, and take <math>\mathfrak{f}(\rho)</math> to be the Artin conductor of this representation. Suppose that, for <math>\rho_0</math> a subrepresentation of <math>\rho \otimes \rho</math> or <math> \rho \otimes \bar{\rho}</math>, <math>L(\rho_0, s)</math> is entire; that is, the Artin conjecture is satisfied for all <math>\rho_0</math>. Take <math>\chi_{\rho}</math> to be the character associated to <math>\rho</math>. Then there is an absolute positive <math>c</math> such that, for <math> x \ge 2</math>, :<math>\sum_{p \,\le\, x,\, p \,\not\mid\, \mathfrak{f}(\rho)} \chi_{\rho}(\text{Fr}_p) \log p = rx + O\biggl(\frac{x^{\beta}}{\beta} + x\exp\biggl(\frac{-c(dn)^{-4} \log x }{3\log \mathfrak{f}(\rho) + \sqrt{\log x}}\biggr) (dn \log (x\mathfrak{f}(\rho))\biggr),</math> where <math>r</math> is <math>1 </math> if <math>\rho</math> is trivial and is otherwise <math>0 </math>, and where <math>\beta</math> is an exceptional real zero of <math>L(\rho, s)</math>; if there is no such zero, the <math>x^{\beta}/\beta</math> term can be ignored. The implicit constant of this expression is absolute.<ref>{{cite book | last1=Iwaniec | first1=Henryk | last2=Kowalski | first2=Emmanuel| title=Analytic Number Theory| year= 2004| location=Providence, RI| publisher=American Mathematical Society| page=111}}</ref>
===Infinite extensions=== The statement of the Chebotarev density theorem can be generalized to the case of an infinite Galois extension <math>L/K </math> that is unramified outside a finite set <math>S </math> of primes of <math>K </math> (i.e. if there is a finite set <math>S </math> of primes of <math>K </math> such that any prime of <math>K </math> not in <math>S </math> is unramified in the extension <math>L/K </math>). In this case, the Galois group <math>G </math> of <math>L/K </math> is a profinite group equipped with the Krull topology. Since <math>G </math> is compact in this topology, there is a unique Haar measure <math>\mu </math> on <math>G </math>. For every prime <math>v </math> of <math>K </math> not in <math>S </math> there is an associated Frobenius conjugacy class <math>F_v </math>. The Chebotarev density theorem in this situation can be stated as follows:<ref name="Section" />
:Let <math>X </math> be a subset of <math>G </math> that is stable under conjugation and whose boundary has Haar measure zero. Then, the set of primes <math>v </math> of <math>K </math> not in <math>S </math> such that <math>F_v \subseteq X </math> has density <math>\mu(X)/\mu(G).</math>
This reduces to the finite case when <math>L/K </math> 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 <math>L </math> are dense in <math>G </math>.
==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 <math>L/K </math> is uniquely determined by the set of primes of <math>K </math> that split completely in it.<ref>Corollary VII.13.10 of Neukirch</ref> A related corollary is that if almost all prime ideals of <math>K </math> split completely in <math>L </math>, then in fact <math>L = K </math>.<ref>Corollary VII.13.7 of Neukirch</ref>
== See also ==
* Splitting of prime ideals in Galois extensions * Grothendieck–Katz p-curvature conjecture
==Notes== <references/>
==References== *{{citation|mr=1395088 | last2= Stevenhagen|first2= P. |last1= Lenstra|first1= H. W. | title = Chebotarëv and his density theorem |journal = The Mathematical Intelligencer |volume=18 |year=1996 | issue= 2|pages=26–37 |doi=10.1007/BF03027290 |url=http://websites.math.leidenuniv.nl/algebra/chebotarev.pdf |citeseerx=10.1.1.116.9409 }} *{{Neukirch_ANT}} *{{Citation | last=Serre | first=Jean-Pierre | author-link=Jean-Pierre Serre | title=Abelian l-adic representations and elliptic curves | orig-year=1968 | year=1998 | publisher=A K Peters, Ltd. | location=Wellesley, MA | edition=Revised reprint of the 1968 original | mr=1484415 | isbn=1-56881-077-6 }} *{{citation |journal=Mathematische Annalen |volume =95|issue= 1 |year=1926|pages= 191–228|doi= 10.1007/BF01206606 |title=Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören |first=N. |last=Tschebotareff}}
Category:Theorems in algebraic number theory Category:Analytic number theory