Editing Chebotarev density theorem

{{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'''<sub>''p''</sub>. 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''<sub>''n''</sub>. 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''<sub>v</sub> 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''<sub>v</sub>. 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''<sub>v</sub> ⊆ 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]]