Mild pro-p-groups and p-extensions of number fields with restricted ramification

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The Galois group of the maximal p-extension of a number field with restricted tame ramification is one of the most mysterious objects in number theory. Little was known about this group until our breakthrough result last year in the case of the rational field which showed that this group was in a large proportional of cases a mild group and hence of cohomological dimension 2. We propose to continue this research in the hope of extending our results by finding an algorithm for mildness. We have obtained such an algorithm in the 4-generator case. We also want to extend our results to other number fields such as quadratic imaginary number fields.