  Let p be an odd prime. We consider the cyclotomic extension T :=
Z_(p)[zeta_{p^2}] of S := Z_(p), with galois group G := (Z/p^2)^*. Since this
extension is wildly ramified, the SG-module T is not projective. We calculate
its cohomology ring H^*(G, T; S), carrying the cup product induced by the ring
structure of T. Proceeding in a somewhat greater generality, our results also
apply to certain Lubin-Tate extensions.
