NOTE SEMINAR TIME: NOON!! Abstract: Let p be a prime number, F a field containing a primitive pth root of unity, and E/F a cyclic extension of degree p, with Galois group G. Let G_E be the absolute Galois group of E. The cohomology groups H^i(E,Fp)=Hî(G_E,Fp) possess a natural structure as FpG-modules and decompose into direct sums of indecomposables. In the 1960s Boreviè and Faddeev gave decompositions of E^*/E^*p -- the case i=1 -- for local fields. We describe the case i=1 for arbitrary fields, and then, using the Bloch-Kato Conjecture, we also determine the case i>1. No small surprise arises from the fact that there exist indecomposable FpG-modules which never appear in these module decompositions. We give several consequences of these results, notably a generalization of the Schreier formula for G_E, connections with Demu¹kin groups, and new families of pro-p-groups that cannot be realized as absolute Galois groups. These results have been obtained in collaboration with D. Benson, J. Labute, N. Lemire, and J. Mináè.