TOPICS
Search

Chebotarev Density Theorem


The Chebotarev density theorem is a complicated theorem in algebraic number theory which yields an asymptotic formula for the density of prime ideals of a number field K that split in a certain way in an algebraic extension L of K. When the base field is the field Q of rational numbers, the theorem becomes much simpler.

Let f(x) be a monic irreducible polynomial of degree n with integer coefficients with root alpha, let K=Q(alpha), let L be the normal closure of K, and let P be a partition (n_1,n_2,...,n_r) of n, i.e., an ordered set of positive integers n_1>=n_2...>=n_r with n=n_1+n_2+....+n_r. A prime is said to be unramified (over the number field K) if it does not divide the discriminant of f. Let S denote the set of unramified primes. Consider the set S_P of unramified primes for which f(x) factors as f_1(x)f_2(x)...f_r(x) modulo p, where f_i is irreducible modulo p and has degree n_i. Also define the density delta(S_P) of primes in S_P as follows:

 delta(S_P)=lim_(N->infty)(#{p in S_P:p<=N})/(#{p in S:p<=N}).

Now consider the Galois group G=Gal(L/Q) of the number field K. Since this is a subgroup of the symmetric group S_n, every element of G can be represented as a permutation of n letters, which in turn has a unique representation as a product of disjoint cycles. Now consider the set of elements G_P of G consisting of disjoint cycles of length n_1, n_2, ..., n_r. Then delta(S_P)=#G_P/#G.

As an example, let f(x)=x^3-2, so K=Q(2^(1/3)) and L=Q(2^(1/3),omega), where omega is a primitive root of unity. Since f has discriminant -108=-2^23^3, the only ramified primes are 2 and 3.

Let p be an unramified prime. Then f has a root (mod p) if and only if 2 has a cube root (mod p), which occurs whenever p=2 (mod 3) or p=1 (mod 3) and 2 has multiplicative order modulo p dividing (p-1)/3. The first case occurs for half of all unramified primes and the second case occurs for one sixth of all primes. In the first case, 2 has a unique cube root modulo p, so f factors as the product of a linear and an irreducible quadratic factor mod p. In the second case, 2 has three distinct cube roots mod p, so f has three linear factors mod p. In the remaining case, which occurs for 1/3 of all unramified primes, f is irreducible mod p. Now consider the corresponding elements of S_3. The first case corresponds to products of 2-cycles and 1-cycles (the identity), of which there are three, or half of the elements of S_3, the second case corresponds to products of three 1-cycles, or the identity, of which there is just one element, or one sixth of the elements of S_3, and the remaining case corresponds to 3-cycles, of which there are two, or one third the elements of S_3. Since Gal(L/Q)=S_3 in this case, the Chebotarev density theorem holds for this example.

The Chebotarev density theorem can often be used to determine the Galois group of a given irreducible polynomial f(x) of degree n. To do so, count the number of unramified primes up to a specified bound for which f factors in a certain way and then compare the results with the fractions of elements of each of the transitive subgroups of S_n with the same cyclic structure. Lenstra provides some good examples of this procedure.


See also

Algebraic Number Theory, Galois Group, Number Field

This entry contributed by David Terr

Explore with Wolfram|Alpha

References

Lenstra, H. "The Chebotarev Density Theorem." http://websites.math.leidenuniv.nl/algebra/Lenstra-Chebotarev.pdf.

Referenced on Wolfram|Alpha

Chebotarev Density Theorem

Cite this as:

Terr, David. "Chebotarev Density Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ChebotarevDensityTheorem.html

Subject classifications