  The power classes of a field are well-known for their ability to parameterize
elementary $p$-abelian Galois extensions. These classical objects have recently
been reexamined through the lens of their Galois module structure. Module
decompositions have been computed in several cases, providing deep new insight
into absolute Galois groups. The surprising result in each case is that there
are far fewer isomorphism types of indecomposables than one would expect
generically, with summands predominately free over associated quotient rings.
Though non-free summands are the exception both in their form and prevalence,
they play the critical role in controlling arithmetic conditions in the field
which allow the rest of the decomposition to be so simple.
  Suppose $m,n \in \mathbb{N}$ and $p$ is prime. In a recent paper, a
surprising and elegant decomposition for $p^m$th power classes has been
computed when the underlying Galois group is a cyclic group of order $p^n$. As
with previous module decompositions, at most one non-free summand appears.
Outside of a particular special case when $p=2$, the structure of this
exceptional summand was determined by a vector $\mathbf{a}\in
\{-\infty,0,\dots,n\}^m$ and a natural number $d$. In this paper we give
field-theoretic interpretations for $\mathbf{a}$ and $d$, showing they are
related to the solvability of a family of Galois embedding problems and the
cyclotomic character associated to $K/F$.
