  Let $\k$ be a global function field in 1-variable over a finite extension of
$\Fp$, $p$ prime, $\infty$ a fixed place of $\k$, and $\A$ the ring of
functions of $\k$ regular outside of $\infty$. Let $E$ be a Drinfeld module or
$T$-module. Then, as in \cite{go1}, one can construct associated characteristic
$p$ $L$-functions based on the classical model of abelian varieties {\it once}
certain auxiliary choices are made. Our purpose in this paper is to show how
the well-known concept of ``maximal separable (over the completion $\k_\infty$)
subfield'' allows one to construct from such $L$-functions certain separable
extensions which are independent of these choices. These fields will then
depend only on the isogeny class of the original $T$-module or Drinfeld module
and $y\in \Zp$, and should presumably be describable in these terms. Moreover,
they give a very useful framework in which to view the ``Riemann hypothesis''
evidence of \cite{w1}, \cite{dv1}, \cite{sh1}. We also establish that an
element which is {\it separably} algebraic over $\k_\infty$ can be realized as
a ``multi-valued operator'' on general $T$-modules. This is very similar to
realizing 1/2 as the multi-valued operator $x\mapsto \sqrt{x}$ on $\C^\ast$.
Simple examples show that this result is false for non-separable elements. This
result may eventually allow a ``two $T$'s'' interpretation of the above
extensions in terms of multi-valued operators on $E$ and certain tensor twists.
