Galois modules appearing as pth-power classes of units of extensions of degree p

We previously described the Galois module structures of pth-power class groups K×/K×p, where K/F is a cyclic extension of degree p over a field F containing a primitive pth root of unity. That description relied upon arithmetic invariants associated with K/F. Here we construct field extensions K/F with prescribed arithmetic invariants, thus completing the classification of Galois modules K×/K×p.