Norm bounds

Hi,

I think I understand how to calculate some class groups. but what worries me is when in an exam we are rounding to calculate size of norms, as we don’t have a square number and are dividing by pi. Is there any guidance on whether we should round up or down in certain certain circumstances. And if say we calculate ideals with norms less than 7 instead of 6 is there anyway we will be able to tell our mistake?

Thanks for your help

Reply:

To be frank, I haven’t yet seen an example where this could cause a problem. But it might be useful to note that an overestimate for the norm bound is preferable to an underestimate, if there is any doubt. You see, with an overestimate, at the worst, you just have to find a few more relations or check a few more ideals for principality that you didn’t really have to. This doesn’t affect the final answer. But if you’ve made an estimate that’s too small, you may in fact have missed entirely a non-principal ideal representing an element in the class group not equivalent to any of those satisfying the (incorrect) bound.

If this paragraph is hard to understand, think about it a little bit. It is important that it make clear sense to you at this point.

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