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| '''Chebotarev's density theorem''' in [[algebraic number theory]] describes statistically the splitting of [[prime number|primes]] in a given [[Galois extension]] ''K'' of the field ''Q'' 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}}.
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| 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
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| :1/''n''
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| 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]]
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| :''Gal''(''K''/''Q''). | |
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| 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
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| :''k''/''n''.
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| == History and motivation == | |
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| When [[Carl Friedrich Gauss]] first introduced the notion of [[gaussian integer|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,
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| : <math> 5 = (1 + 2i)(1-2i) </math> splits completely;
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| : <math> 3 </math> is inert; | |
| : <math> 2 = -i(1+i)^2 </math> ramifies.
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| 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
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| : <math> \Bbb{Z}\subset \Bbb{Z}[i] </math>
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| follows a simple statistical law.
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| 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.
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| 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.
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| == Relation with Dirichlet's theorem ==
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| 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''.
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| ==Formulation==
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| {{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 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''.
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| 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. | |
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| 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>
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| ==Statement==
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| 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
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| :<math>\frac{\#X}{\#G}.</math><ref name="Section">Section I.2.2 of Serre</ref>
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| ===Infinite extensions===
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| 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" />
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| :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>
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| This reduces to the finite case when ''L'' / ''K'' is finite (the Haar measure is then just the counting measure).
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| 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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| ==Important consequences==
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| 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>
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| ==Notes==
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| <references/>
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| ==References==
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| *{{citation|id={{MR|1395088}}
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| | last2= Stevenhagen|first2= P. |last1= Lenstra|first1= H. W.
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| | title = Chebotarëv and his density theorem
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| |journal = The Mathematical Intelligencer
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| |volume=18
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| |year=1996
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| |pages=26–37
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| |doi=10.1007/BF03027290
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| |url=http://websites.math.leidenuniv.nl/algebra/chebotarev.pdf
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| }}
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|
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| *{{Neukirch_ANT}}
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| *{{Citation
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| | last=Serre
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| | first=Jean-Pierre
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| | author-link=Jean-Pierre Serre
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| | title=Abelian l-adic representations and elliptic curves
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| | origyear=1968
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| | year=1998
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| | publisher=A K Peters, Ltd.
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| | location=Wellesley, MA
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| | edition=Revised reprint of the 1968 original
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| | mr=1484415
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| | isbn=1-56881-077-6
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| }}
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| *{{citation
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| |journal=Mathematische Annalen
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| |volume =95|issue= 1 |year=1926|pages= 191–228|doi= 10.1007/BF01206606
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| |title=Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören
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| |first=
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| N. |last=Tschebotareff}}
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| [[Category:Theorems in algebraic number theory]]
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| [[Category:Analytic number theory]]
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