= Adele ring
{title2=$\mathbb A_K$}
{wiki}
For a <number field> $K$, its <adele ring> consists of tuples $(x_v)$ in the completions $K_v$ such that $x_v\in\mathcal O_v$ at all but finitely many finite places. This is the <restricted product> $\mathbb A_K=\prod_v'K_v$, with componentwise operations and the restricted-product topology.
For $K=\mathbb Q$, one has a real component and one <P-adic number> component for each prime. The diagonal image of a <rational number> belongs to the <ring> because only finitely many primes divide its denominator. The construction combines archimedean and congruence information in one locally compact space, enabling global harmonic analysis and formulations of <class field theory>.
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