Editing Chebotarev density theorem (section)

== 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''.