Abstract : Absolute Galois groups of fields are mysterious. One can try to find manageable but still non-trivial quotients of absolute Galois groups. Let p be a prime number. In joint work with D. Benson, N. Lemire, and J. Swallow we consider groups T(E/F) = G_F/\Phi(G_E), where F is a field containing a primitive pth root of unity such that its absolute Galois group G_F is a pro-p group, E/F is a cyclic extension of degree p and \Phi(G_E) is the Frattini subgroup of G_E. We determine all possible groups T(E/F). Further assuming the Bloch-Kato conjecture we determine the \mathbb{F}_p[G_F/G_E]-module H^i(G_E,\mathbb{F}_p) for all i = 1,2,... which extends the previous work of Borevich and Faddeev on the \mathbb{F}_p[G_F/G_E] structure H^1(G_E,\mathbb{F}_p) in the case of local fields.