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In our previous paper we describe the Galois module structures of $p$th-powerclass groups $K^\times/{K^{\times p}}$, where $K/F$ is a cyclic extension ofdegree $p$ over a field $F$ containing a primitive $p$th root of unity. Ourdescription relies upon arithmetic invariants associated with $K/F$. Here weconstruct field extensions $K/F$ with prescribed arithmetic invariants, thuscompleting our classification of Galois modules $K^{\times}/K^{\times p}$

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