Some more classgroups

Hi Sir,

I have been working through the past papers and was hoping you could verify my results for computing class groups:

I get

Q(\sqrt{-30}) to be trivial;

Q(\sqrt{-21}) and Q(\sqrt{35}) to be isomorphic to Z/2 \times Z/2

Also are there any other numbers useful to work out the class groups for.

Many thanks,

(ps I saw the Algebra 3 exam and I thought it was a fair exam so cannot understand why so many people are complaining)

Reply:

The answers I get are

Q(\sqrt{-30}): Z/2\times Z/2

Q(\sqrt{-21}): Z/2\times Z/2

Q(\sqrt{35}):  Z/2

Well, make sure you understand the examples on the coursework and the sample sheet. You can also try computing with any old number you want for practice. To check your answers, I urge everyone to download the program available at

The Pari/GP website

and try to figure out how to use it to compute class groups. You’ll find the command for computing just the class number, i.e., the order of the class group, in the Tutorial. But for the actual group structure, you have to look into the User’s Guide. After downloading, if you need some help, let me know.

Of course this program will not help directly with the exam, since there, you have know *how* it’s computed. But you can use it now for checking your answers.

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