Let $K$ be a number field with $[K:\Rats]=n=r_1+2r_2$ , where $r_1$ denotes the number of real embeddings of $K$ , and $2r_2$ is the number of complex embeddings of $K$ . Let $$\zeta_K(s)$$ be the Dedekind zeta function of $K$ . Also define the following invariants:

1. $h_K$ is the class number, the number of elements in the ideal class group of $K$ .
2. $\operatorname{Reg}_K$ is the regulator of $K$ .
3. $\omega_K$ is the number of roots of unity contained in $K$ .
4. $D_K$ is the discriminant of the extension $K/\Rats$ .

Then:

Note: This is the most general ``class number formula''. In particular cases, for example when $K$ is a cyclotomic extension of $\Rats$ , there are particular and more refined class number formulas.