Given a global field $F$, we can construct the ring of adeles. Given a general locally compact ring $R$, when is it isomorphic to the ring of adeles of some global field $F$ and how can I find $F$ in $R$?



When is a ring the ring of adeles of some global field
Iwasawa gave a characterisation, assuming you are given a subfield F, discrete and such that the quotient is compact. The other conditions are R a semisimple locally compact commutative topological ring with 1 (shared with F). Then R is the adele ring of the global field F.


**Edit**: I believe it is known that you can't get F from knowledge of R alone. I don't remember details or a reference, but it is something like the fact that the Dedekind zeta function doesn't determine the number field? In other words the ramification degrees *e* and residue class extension degrees *f* can be known for each prime, and this will tell you the adele ring R as a restricted product of local fields. But not the field F. Given R, there may be more than one candidate field it contains.


