## Rings of integers

Hi professor,

I was just wandering in the 2007 paper when it says from first principles to determine the ring of algebraic integers in , what this actually means, hope it doesn’t sound like a dumb question. Do we just bear in mind the definition of an algebraic integer and produce a basis for and show that for any element that and are integers?

As regards with the previous question i think the was more like .

Thanks

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Reply:

In essence, yes. A general element of is of the form for . In that problem, you are expected to show that is an algebraic integer if and only if , using just the definition. The context of the problem might help you to understand what is expected from the solution: I noticed at some point that there were students who knew the (important!) formula for the ring of integers in general and could justify it, but then got awfully confused when presented with the same problem for specific .

By the way, to defend my notation , note that there are two different ways to insert brackets, both leading to the same subgroups of . Hence, it’s OK to omit them :-).

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