Class Number / Structure

**Boysilver** · May 25th 2010, 03:47 AM

I've got quite a few practice problems for finding class numbers and structures of imaginary quadratic number fields. For instance, I've just been trying to find the class structure of $\mathbb{Q}(\sqrt{-34})$ and I've got an answer of $C_4$ , the cyclic group of order 4. Is there a reference online that gives the class structure for "small" imaginary quadratic number fields (so that I can check my answers)? I've found that the class numbers are given at, for instance, Tables of small class numbers of imaginary quadratic fields but obviously this doesn't give you all the information. For instance it says that the class number of $\mathbb{Q}(\sqrt{-34})$ is 4 but the class structure could then be either $C_2 \times C_2$ or $C_4$ .

**tonio** · May 25th 2010, 04:13 AM

Originally Posted by Boysilver

I've got quite a few practice problems for finding class numbers and structures of imaginary quadratic number fields. For instance, I've just been trying to find the class structure of $\mathbb{Q}(\sqrt{-34})$ and I've got an answer of $C_4$ , the cyclic group of order 4. Is there a reference online that gives the class structure for "small" imaginary quadratic number fields (so that I can check my answers)? I've found that the class numbers are given at, for instance, Tables of small class numbers of imaginary quadratic fields but obviously this doesn't give you all the information. For instance it says that the class number of $\mathbb{Q}(\sqrt{-34})$ is 4 but the class structure could then be either $C_2 \times C_2$ or $C_4$ .

Well, yes: to find exactly what option it will be you'll have to actually do some work with some ideal generators of ideal group. For this you''ll need Minkowski's theorem and its sequels to bound up the norm for ideals and then take some of them and "play" with them.
I remember in a final exam in graduate schoolw I was given $\mathbb{Q}(\sqrt{105})$ , and while playing with the generators I found they all were of order 2, so the class group is $C_2\times C_2\times C_2$ ...Perhaps in your case it won't be that hard, either

Tonio

**Boysilver** · May 25th 2010, 04:36 AM

Perhaps I didn't make myself clear: I know how to go about finding the class structure, but I want to check my answers. I was just wondering if there was something online that tells you the class group structure of all $\mathbb{Q}(\sqrt{-d})$ for $1 \leq d \leq 100$ , say; as in if anyone knows of a website or book or other reference that just lists (without proof) lots of class groups so that I can check the answers I've ended up with.

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