This question was inspired by
How to prove that the subrings of the rational numbers are noetherian?
which some people found too routine to be of interest. So I have decided to liven things up a bit with the following questions. In the interest of full disclosure, I have not thought seriously about these questions, and I think that I probably could answer at least some of them myself, but I do think they are interesting and, if I may say so, educational.
Find all (commutative!) fields $K$ such that every (unital!) subring $R$ of $K$ is:
a) a principal ideal domain.
b) a Dedekind domain.
c) a Noetherian domain.
I mean here to be asking three different questions, one for each condition. Evidently the classes of such fields are nondecreasing from a) to b) and from b) to c).
If you would like to answer the question with a), b) or c) replaced by some other standard property of commutative rings — especially if it yields a different class of fields than in the first three questions — please feel free.
Addendum: How about
d) a Dedekind domain if it is integrally closed?
e) a PID if it is integrally closed?
Best Answer
Regarding question (c), I can tell you exactly which integral domains have only Noetherian subrings by quoting the aptly titled Integral domains with Noetherian subrings by Robert Gilmer:
If $K$ is the field of fractions and $\operatorname{char}(K)=0$, we just need $[K:\mathbb{Q}]<\infty$.
If $\operatorname{char}(K)=p$ with prime subfield $k$, we need $K$ to be either finite or a finite algebraic extension of a $k[X]$ for some transcendental $X$.
I guess this pretty much restricts the answers to questions (a) and (b)….