  The ad\`ele ring $\mathbb A_K$ of a global field $K$ is a locally compact,
metrizable topological ring which is complete with respect to any invariant
metric on $\mathbb A_K$. For a fixed global field $F$ and a possibly infinite
algebraic extension $E/F$, there is a natural partial ordering on $\{\mathbb
A_K:F\subseteq K\subseteq E\}$. Therefore, we may form the direct limit \[
  \mathbb A_E = \varinjlim \mathbb A_K \] which provides one possible
generalization of ad\`ele rings to arbitrary algebraic extensions $E/F$. In the
case where $E/F$ is Galois, we define an alternate generalization of the
ad\`eles, denoted $\bar{\mathbb V}_E$, to be a certain metrizable topological
ring of continuous functions on the set of places of $E$. We show that
$\bar{\mathbb V}_E$ is isomorphic to the completion of $\mathbb A_E$ with
respect to any invariant metric and use this isomorphism to establish several
topological properties of $\mathbb A_E$.
