## 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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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 :-).