On cyclic fields of odd prime degree p with infinite Hilbert p-class field towers

Let k be a cyclic extension of odd prime degree p of the field of rational numbers. If t denotes the number of primes that ramify in k, it is known that the Hilbert p-class field tower of k is infinite if t > 3 + 2rootp. For each t > 2 + rootp, this paper shows that a positive proportion of such fields k have infinite Hilbert p-class field towers.