Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is there any sort of a bound for the magnitude of this residue? I've been looking at some algebraic number theory problems from Princeton's general exam. One of them is the following:

Why does a ''large'' fundamental unit suggest a ''small'' class number and vice versa?

To me it seems like this has to have something to do with the relation given by the residue of the zeta function. A ''large'' fundamental unit would imply a large regulator, but I can't see why this would force the class number to be small unless we can bound the magnitude of the residue. Any ideas?

share|improve this question
2  
There are a great many zeta-functions in mathematics today. I think you are referring to the zeta-function of a number field, the Dedekind zeta-function, but perhaps you could edit your question to include this information explicitly. –  Gerry Myerson Nov 5 '11 at 2:49

1 Answer 1

up vote 8 down vote accepted

As you anticipated, by estimating the value of $L$-functions at $s=1$, one can obtain bounds on the class group and the regulator. Roughly, one has an inequality $h R < |D|^{1/2 + \epsilon}$, where $h$, $R$, and $D$ are the class number, regulator, and discriminant of the field respectively. More precisely, one has the Brauer-Siegel theorem.

share|improve this answer
    
The link didn't seem to work for some reason - please feel free to edit the answer so it works, I don't know how. –  Splice Nov 5 '11 at 4:54
1  
There are some characters in urls, like the long dash here, that MSE gets in a tizzy over. WP redirects it though if you replace with a normal dash. –  anon Nov 5 '11 at 5:07

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.