Class field towers of imaginary quadratic fields (Q1087584)

Let \(F\) be an imaginary quadratic field whose ideal class group has an elementary abelian Sylow 3-subgroup of order \(3^ 2\). \textit{A. Scholz} and \textit{O. Taussky} [J. Reine Angew. Math. 171, 19--41 (1934; Zbl 0009.10202)] proved that if \(F\) has a certain capitulation type then the 3-class field tower of \(F\) is of length 2. The present authors prove this in a simpler way. For another particular capitulation type they show, by constructing a group theoretic example, that it is possible for the 3-class field tower to be of length 3. This corrects an assertion made by Scholz and Taussky [op. cit.].