A note on the 3-class field tower of a cyclic cubic field. (Q2641977)

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.