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Apr 23, 2010 at 18:13 comment added Franz Lemmermeyer I'm not sure I understand your question; what I meant to say was that the p-tower is finite if the class groups are "small", and that it is infinite if they are "large", and that neither group theoretical calculations nor improvements of Golod-Shafarevich tell us what to expect in the cases in between. Personally I fear that there is no nice and simple result that separates the finite towers from the infinite.
Apr 23, 2010 at 14:49 comment added Cam McLeman I'm intrigued by the last comment -- "where no one really knows what is happening." Not that I'm disagreeing, but did you have something specific in mind here?
Apr 23, 2010 at 8:19 history answered Franz Lemmermeyer CC BY-SA 2.5