llllllllllllllllllll

". . .if you fix p, and study the fields K_n obtained by adjoining a p^n-th root of unity to Q, then I believe that the exponent of p in the class number is independent of n (at least for n large enough). . ."

The correct (but still vague) statement here would be, not that the p part of the class numbers are independent of n for large n, but that the growth of the class numbers can be described very explicitly in terms of n. Roughly speaking: the p-part of the class number of K_n has exponent mp^n+ln+v for some integers m, l, v.