1.Let $p$ be an odd prime number. Is there a classification of cyclotomic fields of the form $\mathbb{Q}(\zeta_p)$ with class number 1?
2.Is there such a classification for general cyclotomic fields $\mathbb{Q}(\zeta_m)$, where $m$ is a positive integer?
1 Answer
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The complete list of $n$ for which ${\bf Q}(\zeta_n)$ has unique factorization is 1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90.
Source; https://en.wikipedia.org/wiki/Cyclotomic_field#List_of_class_numbers_of_cyclotomic_fields

$\begingroup$ Thanks. So, it is in the Washington's book. $\endgroup$ Commented Jun 26, 2021 at 12:42

1$\begingroup$ The result is due to Montgomery, Hugh L. and Masley, J.Myron. "Cyclotomic fields with unique factorization." Journal für die reine und angewandte Mathematik, vol. 1976, no. 286287, 1976, pp. 248256. doi.org/10.1515/crll.1976.286287.248 $\endgroup$ Commented Jun 26, 2021 at 23:51