Question

Let $p$ be a prime number and $F$ a finite extension of ${\mathbf Q}_p$ or of ${\mathbf F}_p((t))$. I'm going to define a natural map from the set ${\mathcal S}_p(F)$ of degree-$p$ separable extensions of $F$ into the group $F^\times/F^{\times p-1}$, and I want to know if this map has a name, or if you have come across it somewhere.

I need the auxillary field $K=F(\root{p-1}\of{F^\times})$, with group $G={\rm Gal}(K|F)$. Note that ${\rm Hom}(G,{\mathbf F}_p^\times)$ is canonically isomorphic to $F^\times/F^{\times p-1}$. Note also that $K$ contains a primitive $p$-th root of $1$ if $F$ has characteristic $0$.

It can be shown that for every $E\in{\mathcal S}_p(F)$, the compositum $EK$ is cyclic of degree $p$ over $K$ and galoisian over $F$. Therefore there is a $G$-stable ${\mathbf F}_p$-line $D\subset K^\times/K^{\times p}$ or $D\subset K^+/\wp(K^+)$, in characteristic $0$ and $p$ respectively, such that $EK=K(\root p\of D)$ or $EK=K(\wp^{-1}(D))$. (Here $\wp$ is the endomorphism $x\mapsto x^p-x$ of the ${\mathbf F}_p$-space $K^+$ in characteristic $p$.)

Let $\chi:G\to{\mathbf F}_p^\times$ be the character through which $G$ acts on $D$. This assignment $E\mapsto\chi$ can be thought of as a map ${\mathcal S}_p(F)\to F^\times/F^{\times p-1}$. I would like to know : Does this map manifest itself naturally in some context ? Does it have a name ?

Addendum (2011/11/01) The reason why I was interested in this map ${\mathcal S}_p(F)\to F^\times/F^{\times p-1}$ is that it partitions the set ${\mathcal S}_p(F)$ into $(p-1)^2=|F^\times/F^{\times p-1}|$ parts and one can compute the contribution of each of these parts to Serre's degree-$p$ mass formula. This allows you to compute the contribution of cyclic extensions, or of those whose galoisian closure has a given group as group of automorphisms, or has ramification properties given in advance. See arXiv:1110.6702 for the details.

Thanksgiving (2011/11/30) I should have mentioned that the fact that for every $E\in{\mathcal S}_p(F)$, the compositum $EK$ is cyclic of degree $p$ over $K$ and galoisian over $F$ is a direct consequence of "Galois's Last Theorem" characterising solvable transitive subgroups of $\mathfrak{S}_p$ (see Solvable transitive groups of prime degree).