ac.commutative algebra - For which fields K is every subring of K...?

Let me put together the previous two answers (plus epsilon) to give an answer to all three questions.

Step 1: By Gilmer's theorem, a field $K$ has all its subrings Noetherian iff:

(i) It is a finite extension of $mathbb{Q}$, or
(ii) It is an algebraic extension of $mathbb{F}_p$ or a finite extension of $mathbb{F}_p(t)$.

Step 2: Suppose $K$ is a number field which is not $mathbb{Q}$. We may write $K = mathbb{Q}[alpha]$ for some algebraic integer $alpha$. Then $R = mathbb{Z}[2alpha]$ is a non-integrally closed subring of $K$ so is not a Dedekind domain. So the only field of characteristic $0$ which has every subring a Dedekind domain is $mathbb{Q}$, in which case (by the previous question) every subring is a PID.

Step 3: Suppose $K$ has characteristic $p > 0$. If $K$ is algebraic over $mathbb{F}_p$, then every subring is a field, hence also Dedekind and a PID. If $K$ is a finite extension of $mathbb{F}_p(t)$ then it admits a subring of the form $mathbb{F}_p[t^2,t^3]$, which is not integrally closed.

So the fields for which every subring is a Dedekind ring are $mathbb{Q}$ and the algebraic extensions of $mathbb{F}_p$. For all such fields, every subring is in fact a PID.