The adèle ring of a global field is a locally compact, metrizable topological ring which is complete with respect to any invariant metric on . For a fixed global field and a possibly infinite algebraic extension , there is a natural partial ordering on . Therefore, we may form the direct limit
which provides one possible generalization of adèle rings to arbitrary algebraic extensions . In the case where is Galois, we define an alternate generalization of the adèles, denoted by , to be a certain metrizable topological ring of continuous functions on the set of places of . We show that is isomorphic to the completion of with respect to any invariant metric and use this isomorphism to establish several topological properties of .
"Direct limits of adèle rings and their completions." Rocky Mountain J. Math. 50 (3) 1021 - 1043, June 2020. https://doi.org/10.1216/rmj.2020.50.1021