On a theorem of Koch (Q1358966)

The author studies the question of which quadratic number fields have infinite 2-class field tower. He strengthens a theorem of the reviewer [cf. \textit{H. Koch}, J. Reine Angew. Math. 214/215, 201-206 (1964; Zbl 0123.03904)] about such fields with class group containing a subgroup of type (4, 4, 4). His method of proof is different from the method of the reviewer. He uses a theorem of \textit{J. Martinet} [Invent. Math. 44, 65-73 (1978; Zbl 0369.12007)] about infinite 2-class towers for fields of higher degree. Furthermore, the author gives some new interesting examples of infinite class field towers.