General Statement of The Class Number Formula
Let K be a number field with =n=r1+2r2, where denotes the number of real embeddings of K, and is the number of complex embeddings of K. Let
be the Dedekind zeta function of K. Also define the following invariants:
- is the class number, the number of elements in the ideal class group of K.
- is the regulator of K.
- is the number of roots of unity contained in K.
- is the discriminant of the extension K/Q.
Then:
Theorem 1 (Class Number Formula) The Dedekind zeta function of K, converges absolutely for 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.
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