5 Some additive galois cohomology rings

Let p ≥ 3 be a prime. We consider the cyclotomic extension Z (p) [ζ p 2 ] | Z (p) , with galois group G = (Z/p 2) *. Since this extension is wildly ramified, the Z (p) G-module Z (p) [ζ p 2 ] is not projective. We calculate its cohomology ring H * (G, Z (p) [ζ p 2 ]; Z (p)), carrying the cup product induced by the ring structure of Z (p) [ζ p 2 ]. Formulated in a somewhat greater generality, our results also apply to certain Lubin-Tate extensions.