Editing Chebotarev density theorem (section)

== 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.