A note on the 3-class field tower of a cyclic cubic field. (English) Zbl 1123.11036

In this note the author proves the following theorem. Let \(K\) be a cyclic cubic number field such that exactly three rational primes ramify in \(K/Q\). Then the 3-class field tower of \(K\) is greater than 1 if and only if the class number of the genus field of \(K\) is divisible by 3. To prove this theorem, the author makes use of a result from his earlier work, Chebotarev’s monodromy theorem, and GAP-groups.


11R29 Class numbers, class groups, discriminants
11R16 Cubic and quartic extensions
11R37 Class field theory


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