We consider projective, irreducible, non-singular curves over an algebraically closed field \(\Bbbk \) . A cover \(Y \rightarrow X\) of such curves corresponds to an extension \(\Omega /\Sigma \) of their function fields and yields an isomorphism \(\mathbb {A}_{Y} \simeq \mathbb {A}_{X} \otimes _{\Sigma } \Omega \) of their geometric adele rings. The primitive element theorem shows that \(\mathbb {A}_{Y}\) is a quotient of \(\mathbb {A}_{X}[T]\) by a polynomial. In general, we may look at quotient algebras \(\mathbb {A}_{X}\lbrace \hspace{0.0pt}{\mathbbm {p}}\hspace{0.0pt}\rbrace = \mathbb {A}_{X}[T]/(\mathbbm {p}(T))\) where \(\mathbbm {p}(T) \in \mathbb {A}_{X}[T]\) is monic and separable over \(\mathbb {A}_{X}\) , and try to characterize the field extensions \(\Omega /\Sigma \) lying in \(\mathbb {A}_{X}\lbrace \hspace{0.0pt}{\mathbbm {p}}\hspace{0.0pt}\rbrace \) which arise from covers as above. We achieve this in two ways; the first, topologically, as those \(\Omega \) which embed discretely in \(\mathbb {A}_{X}\lbrace \hspace{0.0pt}{\mathbbm {p}}\hspace{0.0pt}\rbrace \) . The second is the characterization of such subfields \(\Omega \) as those which satisfy the additive analog of the product formula in classical adele rings. The technical machinery is based on the use of Tate topologies on the quotient algebras \(\mathbb {A}_{X}\lbrace \hspace{0.0pt}{\mathbbm {p}}\hspace{0.0pt}\rbrace \) . These are not locally compact, but we are able to define an additive content function as an index measuring the discrepancy of dimensions in commensurable subspaces.
