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

p-Tower Groups over Quadratic Imaginary Number Fields