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class number formula : ウィキペディア英語版
class number formula
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
==General statement of the class number formula==
We start with the following data:
* is a number field.
* , where denotes the number of real embeddings of , and is the number of complex embeddings of .
* is the Dedekind zeta function of .
* is the class number, the number of elements in the ideal class group of .
* is the regulator of .
* is the number of roots of unity contained in .
* is the discriminant of the extension .
Then:
:Theorem (Class Number Formula). converges absolutely for and extends to a meromorphic function defined for all complex with only one simple pole at , with residue
:: \lim_ (s-1) \zeta_K(s) = \frac \cdot h_K \cdot \operatorname_K}{w_K \cdot \sqrt{|D_K|}}
This is the most general "class number formula". In particular cases, for example when is a cyclotomic extension of , there are particular and more refined class number formulas.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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