Construction and classification of some Galois modules

In our previous paper we describe the Galois module structures of $p$th-power class groups $K^{\times }\mathrm{/}{K}^{\times p}$, where $K\mathrm{/}F$ is a cyclic extension of degree $p$ over a field $F$ containing a primitive $p$th root of unity. Our description relies upon arithmetic invariants associated with $K\mathrm{/}F$. Here we construct field extensions $K\mathrm{/}F$ with prescribed arithmetic invariants, thus completing our classification of Galois modules $K^{\times }\mathrm{/}{K}^{\times p}$.