Formalizing the Ring of Adèles of a Global Field

Abstract: The ring of ad`eles of a global field and its group of units, the group of id`eles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we formalized adic valuations on Dedekind domains. We present some applications, including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its id`ele class group.