The frattini subgroup of the absolute Galois group of a local field

Let K be a local field, T the maximal tamely ramified extension of K, F the fixed field in K _sof the Frattini subgroup of G(K), and J the compositum of all minimal Galois extensions of K containing T. The main result of the paper is that F=J. If K is a global field and K _solv is the maximal prosolvable extension of K, then the Frattini group of {Mathematical expression}% MathType!End!2!1!(K _solv/K) is trivial.