Realising the cup-product of local Tate duality

We present an explicit description, in terms of central simple algebras, of a cup-product map which occurs in the statement of local Tate duality for Galois modules of prime order p. Given cocycles f and g, we construct a central simple algebra of dimension p^2 whose class in the Brauer group gives the cup-product of f with g. This algebra is as small as possible.