Galois module structure of Galois cohomology for embeddable cyclic extensions of degree p^n

Let p>2 be prime, and let n,m be positive integers. For cyclic field extensions E/F of degree p^n that contain a primitive pth root of unity, we show that the associated F_p[Gal(E/F)]-modules H^m(G_E,mu_p) have a sparse decomposition. When E/F is additionally a subextension of a cyclic, degree p^{n+1} extension E'/F, we give a more refined F_p[Gal(E/F)]-decomposition of H^m(G_E,mu_p).