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?
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1 Answer
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2
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
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$\begingroup$ Thanks. So, it is in the Washington's book. $\endgroup$Ehsan Shahoseini– Ehsan Shahoseini2021-06-26 12:42:58 +00:00Commented Jun 26, 2021 at 12:42
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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. 286-287, 1976, pp. 248-256. doi.org/10.1515/crll.1976.286-287.248 $\endgroup$Gerry Myerson– Gerry Myerson2021-06-26 23:51:56 +00:00Commented Jun 26, 2021 at 23:51