  The modern theory of class field towers has its origins in the study of the
p-class field tower over a quadratic imaginary number field, so it is fitting
that this problem be the first in the discipline to be nearing a solution. We
survey the state of the subject and present a new cohomological condition for a
quadratic imaginary number field to have an infinite p-class field tower (for p
odd). Under an additional hypothesis, we refine this to a necessary and
sufficient condition and describe an algorithm for evaluating this condition
for a given quadratic imaginary number field.
