Chebotarev's Density Theorem - History and Motivation

History and Motivation

When Carl Friedrich Gauss first introduced the notion of complex integers Z, 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,

splits completely;

is inert;

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. 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

follows a simple statistical law.

Similar statistical laws also hold for splitting of primes in the 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, 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.