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 focusing on fields of small degree (n=3 and n=4), what about a less restrictive question: can we find two such fields that have the same regulator (no discriminant restrictions)?



Number fields with same discriminant and regulator?
Yes, see e.g. the paper "Arithmetically equivalent number fields of small degree" (Google for it) by Bosma and de Smit. 


In brief: two number fields $K$ and $K'$ are said to be **arithmetically equivalent** if they have the same Dedekind zeta function. A famous group-theoretic construction of Perlis (Journal of Number Theory, 1977) gives many nontrivial (i.e., non-isomorphic) pairs of arithmetically equivalent number fields. Remarkably, this construction works equally well to construct isospectral, non-isometric Riemannian manifolds, as was later shown by Sunada.


Arithmetically equivalent number fields necessarily share many of the simplest invariants, for instance they have equal discriminants. 


As the aformentioned paper explains, for arithmetically equivalent $K$ and $K'$, comparing zeta functions gives


$h(K)r(K) = h(K')r(K')$,


where $h$ is the class number and $r$ is the regulator. Therefore, to get an affirmative answer to your question you want a nontrivial pair of arithmetically equivalent number fields $K$ and $K'$ with $h(K) = h(K')$. The paper by Bosma and de Smit gives such examples.


