Chebotarev density theorem - Revision history

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	== Relation with Dirichlet's 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 ''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 [[modular arithmetic\|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''.		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 [[modular arithmetic\|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==		==Formulation==
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	where the constant implied in the [[big-O notation]] is absolute, ''n'' is the degree of ''L'' over '''Q''', and Δ its discriminant.		where the constant implied in the [[big-O notation]] is absolute, ''n'' is the degree of ''L'' over '''Q''', and Δ its discriminant.

	The effective form of the Chebotarev density theory becomes much weaker without GRH. Take ''L'' to be a finite Galois extension of ''Q'' with Galois group ''G'' and degree ''d''. Take <math>\rho</math> to be a nontrivial irreducible representation of ''G'' of degree ''n'', 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>,		The effective form of the Chebotarev density theory becomes much weaker without GRH. Take ''L'' to be a finite Galois extension of ''Q'' with Galois group ''G'' and degree ''d''. Take <math>\rho</math> to be a nontrivial irreducible representation of ''G'' of degree ''n'', 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>		:<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 1 if <math>\rho</math> is trivial and is otherwise 0, and where <math>\beta</math> is an [[Siegel zero\|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>		where <math>r</math> is 1 if <math>\rho</math> is trivial and is otherwise 0, and where <math>\beta</math> is an [[Siegel zero\|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>
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	::<math>\frac{\mu(X)}{\mu(G)}.</math>		::<math>\frac{\mu(X)}{\mu(G)}.</math>

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

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2025-05-03T18:09:00Z

Effective version

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{{short description|Describes statistically the splitting of primes in a given Galois extension of Q}}
The '''Chebotarev density theorem''' in [[algebraic number theory]] describes statistically the splitting of [[prime number|primes]] in a given [[Galois extension]] ''K'' of the field <math>\mathbb{Q}</math> of [[rational number]]s. Generally speaking, a prime integer will factor into several [[Ideal number|ideal primes]] in the ring of [[algebraic integer]]s 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 {{harv|Tschebotareff|1926}}.

A special case that is easier to state says that if ''K'' is an [[algebraic number field]] which is a Galois extension of <math>\mathbb{Q}</math> 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 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 [[gaussian integer|complex integers]] ''Z''[{{itco|''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 {{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 1/2, and likewise for the primes that remain primes in ''Z''[{{itco|''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

: <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 field|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 extension|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 [[modular arithmetic|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==

In their survey article, {{harvtxt|Lenstra|Stevenhagen|1996}} 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 of an integer|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 conjugate]]s, ''G'' is [[Faithful representation|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.<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 ''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
:<math>\frac{\#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 results in number theory|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 ''L''/''K'' is a finite Galois extension with Galois group ''G'', and ''C'' a union of conjugacy classes of ''G'', the number of unramified primes of ''K'' of norm below ''x'' with Frobenius conjugacy class in ''C'' 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, ''n'' is the degree of ''L'' over '''Q''', and Δ its discriminant.

The effective form of the Chebotarev density theory becomes much weaker without GRH. Take ''L'' to be a finite Galois extension of ''Q'' with Galois group ''G'' and degree ''d''. Take <math>\rho</math> to be a nontrivial irreducible representation of ''G'' of degree ''n'', 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 1 if <math>\rho</math> is trivial and is otherwise 0, and where <math>\beta</math> is an [[Siegel zero|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 ''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:<ref name="Section" />

: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
::<math>\frac{\mu(X)}{\mu(G)}.</math>

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.<ref>Corollary VII.13.10 of Neukirch</ref> A related corollary is that if almost all prime ideals of ''K'' split completely in ''L'', then in fact ''L'' = ''K''.<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]]