Are there non-isomorphic number fields (say of the same degree and signature) that have the same discriminant and regulator? I'm guessing the answer is no - why?
And are therefocusing on fields of small degree (n=3 and n=4), what about a less restrictive question: can we find two such fields that just have the same regulator (no discriminant restrictions)? I'm guessing the answer is yes - can't think of an example.