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 Q[(103)^{1/2}], 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 Q[(103)^{1/2}] and show that for any element a + b[(103)^{1/2}] that a and b are integers?

As regards with the previous question i think the nZ^d was more like n(Z^d).
Thanks
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Reply:

In essence, yes. A general element of Q[(103)^{1/2}] is of the form a+b(103)^{1/2} for a,b\in Q. In that problem, you are expected to show that a+b(103)^{1/2} is an algebraic integer if and only if a,b\in Z, 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 Q[\sqrt{d}] and could justify it, but then got awfully confused when presented with the same problem for specific d.

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

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