Editing Chebotarev density theorem (section)

==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>