Published online by Cambridge University Press: 17 April 2009
Let k be an imaginary quadratic number field and let k1 be the 2-Hilbert class field of k. If Ck,2, the 2-Sylow subgroup of the ideal class group of k, is elementary and |Ck,2|≥ 8, we show that Ck1,2 is not cyclic. If Ck,2 is isomorphic to Z/2Z × Z/4Z and Ck1,2 is elementary we show that k has finite 2-class field tower of length at most 2.