  In this paper, we describe the Galois module structure of
$J=\mathbf{K}^{\times}/\mathbf{K}^{\times p}$, where $\mathbf{K}$ is an
extension of a local field $\mathbf{k}$ containing a primitive $p$-th root of
unity: for instance, if $\mathbf{K}/\mathbf{k}$ is a $p$-elementary abelian
extension, we prove that $J$ is a module of constant Jordan type, with stable
Jordan type $[1]^2$, which, in a way, extends the result of J. Min\'a\v{c} and
J. Swallow. Also, we take profit from our proof by computing some invariants,
which were previously introduced by A. Adem, W. Gao, D. B. Karageuzian and J.
Min\'a\v{c} only for $p=2$.
