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Concept
Class number formula
Summary
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
We start with the following data:
K is a number field.
[K : Q] = n = r1 + 2r2, where r1 denotes the number of real embeddings of K, and 2r2 is the number of complex embeddings of K.
ζK(s) is the Dedekind zeta function of K.
hK is the class number, the number of elements in the ideal class group of K.
RegK is the regulator of K.
wK is the number of roots of unity contained in K.
DK is the discriminant of the extension K/Q.
Then:
Theorem (Class Number Formula). ζK(s) converges absolutely for Re(s) > 1 and extends to a meromorphic function defined for all complex s with only one simple pole at s = 1, with residue
This is the most general "class number formula". In particular cases, for example when K is a cyclotomic extension of Q, there are particular and more refined class number formulas.
The idea of the proof of the class number formula is most easily seen when K = Q(i). In this case, the ring of integers in K is the Gaussian integers.
An elementary manipulation shows that the residue of the Dedekind zeta function at s = 1 is the average of the coefficients of the Dirichlet series representation of the Dedekind zeta function. The n-th coefficient of the Dirichlet series is essentially the number of representations of n as a sum of two squares of nonnegative integers. So one can compute the residue of the Dedekind zeta function at s = 1 by computing the average number of representations. As in the article on the Gauss circle problem, one can compute this by approximating the number of lattice points inside of a quarter circle centered at the origin, concluding that the residue is one quarter of pi.
The proof when K is an arbitrary imaginary quadratic number field is very similar.
In the general case, by Dirichlet's unit theorem, the group of units in the ring of integers of K is infinite.
Official source
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