I'm studying the number field sieve factorization method, but I am having some trouble with the definition of number ring. I have found two different definitions:
- A number ring is a subring of a number field.
- A number ring is the subring of algebraic integers of some field.
They should not be equivalent, for example if we take the number field $\mathbb{Q}[\sqrt{5}]$, then for the first definition $\mathbb{Z}[\sqrt{5}]$ is a number ring, but for the second definition, the number ring is $\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$.
I'm pretty sure that with the second definition we have that a number ring is a Dedekind domain. But what happens with the first definition?
Thanks for your help.
The first definition is OK, but not much can be said about number rings.
The second definition corresponds to what is usually the ‘ring of integers of the number field’, and in my opinion, it's better to use this terminology,to prevent any ambiguity.
To answer your question, if the number ring doesn't satisfy the second definition, it can't be a Dedekind domain since it is not integrally closed.