Consider the following two problems of Number Theory.
For which prime $p, \Bbb Z[\omega]$ is a UFD, where $\omega$ is a $p$-th root of unity?
Does Unique Factorization occur for infinitely many $d>0$ in the ring of integers of $\Bbb Q\bigl(\sqrt{d}\mkern2mu\bigr)$?
Both these problems are not yet completely solved. My question is
As both of the problems are concerned with unique factorization in a ring, is there any relation between these two problems. Does solution of one imply the solution of other? Also, what are the best-known results on these problems?
It is a UFD for all primes $p\le 19$ and never again for any prime $p>19$.
The class number is conjecturally $1$ for infinitely many squarefree $d>0$ (Gauß), i.e., the ring of integers is a UFD for infinitely many $d>0$. This is still open.
Conclusion: The two statements are essentially not related. However, they are both part of the "class number one problem".
Remark to 1: The class numbers of $\Bbb Q(\zeta_p)$ grow rapidly with $p$: \begin{array}{cc} p & h_{\Bbb Q(\zeta_p)} \\ \hline 2 & 1 \\ 3 & 1 \\ 5 & 1 \\ 7 & 1 \\ 11 & 1 \\ 13 & 1 \\ 17 & 1 \\ 19 & 1 \\ 23 & 3 \\ 29 & 8 \\ 31 & 9 \\ 37 & 37 \\ 41 & 121 \\ 43 & 211 \\ 47 & 695 \\ 53 & 4889 \\ 59 & 41241 \\ 61 & 76301 \\ 67 & 853513 \\ 71 & 3882809 \\ 73 & 11957417 \\ 79 & 100146415 \\ 83 & 838216959 \\ 89 & 13379363737 \\ 97 & 411322824001 \\ 101 & 3547404378125 \\ 103 & 9069094643165 \\ 107 & 63434933542623 \\ 109 & 161784800122409 \\ 113 & 1612072001362952 \\ 127 & 2604529186263992195 \\ 131 & 28496379729272136525 \\ 137 & 646901570175200968153 \\ 139 & 1753848916484925681747 \\ 149 & 687887859687174720123201 \end{array}