Given an algebraic extension $K/\mathbb Q$ define the ring of integers $\mathcal O_K$ of $K$ as the integral closure of $\mathbb Z$ inside $K$. My question is the following: If $K/\mathbb Q$ is infinite, can $\mathcal O_K$ be a UFD?
Some ideas: Inspired from the answer of user KCd in this MSE post I think it is enough to find a tower of extensions $K_1\subset K_2\subset\cdots$ such that at the $k$th stage all the primes above the first $k$ rational primes are principal and remain prime in all further stages, for all $k=1,2,\dots$ Assuming this is the case and putting $K=\bigcup_i K_i$, then all the prime ideals of $\mathcal O_K$ are principal, in particular $\mathcal O_K$ is noetherian. Since $\mathcal O_K$ is integrally closed and of Krull dimension $1$ (integral extensions preserve Krull dimension), this implies that $\mathcal O_K$ is a Dedekind domain, but the prime ideals are principal and thus $\mathcal O_K$ is a PID (which is stronger than being a UFD).
Assume inductively that the $k$th stage has been constructed. To construct the next stage we need to find an extension $K_{k+1}/K_k$ such that (a) all the primes $\mathfrak p_1,\dots, \mathfrak p_n$ above the first $k$ rational primes remain prime and (b) the primes $\mathfrak q_1,\dots, \mathfrak q_m$ above the $k+1$-st rational prime factor as products of principal prime ideals. For (a), it is enough to make sure that the extension is defined by a monic polynomial with integral coefficients that is irreducible mod $\mathfrak p_i$, for $i=1,\dots, n$. (b) seems to me more difficult, as it has to do with global properties, and a local argument may not suffice. In any case, this attempt induces the following question:
Follow up question: Let $K$ be a number field. For any choice of (distinct) primes $\mathfrak p_1,\dots, \mathfrak p_n, \mathfrak q_1,\dots, \mathfrak q_m$ of $\mathcal O_K$, does there exist a finite extension $L/K$ such that $\mathfrak p_1,\dots, \mathfrak p_n$ remain prime and $\mathfrak q_1,\dots, \mathfrak q_m$ factor into products of principal prime ideals?
