Finite quotients of Galois pro- p groups and rigid fields

For a prime number p, the author shows that if two certain canonical finite quotients of a finitely generated Bloch–Kato pro-p group G coincide, then G has a very simple structure, i.e., G is a p-adic analytic pro-p group (see Theorem 1). This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions of a field F—with F containing a primitive p-th root of unity—coincide, then F is p-rigid (see Corollary 1). The proof relies only on group-theoretic tools, and on certain properties of Bloch–Kato pro-p groups.