show · nf.class_number_formula all knowls · up · search:

If $K$ is a number field with signature $(r_1, r_2)$, discriminant $D$, regulator $R$, class number $h$, containing $w$ roots of unity, and Dedekind $\zeta$-function $\zeta_K$, then $\zeta_K$ has a meromorphic continuation to the whole complex plane with a single pole at $s=1$, which is of order $1$. The analytic class number formula gives the residue at this pole: $$\lim_{s\to 1}\ (s-1)\zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2} R\cdot h}{w\sqrt{|D|}} .$$

Authors:
Knowl status:
• Review status: reviewed
• Last edited by John Jones on 2020-10-12 09:26:47
Referred to by:
History:
Differences