On the T-ramified, S-split p-class field towers over an extension of degree prime to p

Let K be a number field, p a prime, and let be the T-ramified, S-split p-class field tower of K, i.e., the maximal pro-p-extension of K unramified outside T and totally split on S, where T and S are disjoint finite sets of places of K. Using a theorem of Tate on nilpotent quotient groups, we give (Theorem 2 in Section 3) an elementary characterisation of the finite extensions L/K, with a normal closure of degree prime to p, such that the analogous p-class field tower of L is equal to the compositum . This N.S.C. only depends on classes and units of L. Some applications and examples are given.