primes are coprime, follows. Remark. is not uniquely divisible. Let and be given. Then both satisfy the equation and clearly ( is well-defined, because only finitely many primes divide ). In this case, being uniquely divisible is equivalent to being torsion-free, which is not true for since but and Remark. The fourth statement is a special case of the strong approximation theorem. Haar measure on the adele ring Definition. A function is called simple if where are measurable and for almost all Theorem. Since is a locally compact group with addition, there is an additive Haar measure on This measure can be normalized such that every integrable simple function satisfies: where for is the measure on such that has unit measure and is the Lebesgue measure. The product is finite, i.e., almost all factors are equal to one. The idele group Definition. Define the idele group of as the group of units of the adele ring of that is The elements of the idele group are called the ideles of Remark. is equipped with a topology so that it becomes a topological group. The subset topology inherited from is not a suitable candidate since the group of units of a topological ring equipped with subset topology may not be a topological group. For example, the inverse map in is not continuous. The sequence converges to To see this let be neighbourhood of without loss of generality it can be assumed: Since for all for large enough. However, as was seen above the inverse of this sequence does not converge in Lemma. Let be a topological ring. Define: Equipped with the topology induced from the product on topology on and is a topological group and the inclusion map is continuous. It is the coarsest topology, emerging from the topology on that makes a topological group. Proof. Since is a topological ring, it is sufficient to show that the inverse map is continuous. Let be open, then is open. It is necessary to show is open or equivalently, that is open. But this is the condition above. The idele group is equipped with the topology defined in the Lemma making it a topological group. Definition. For a subset of places of set: Lemma. The following identities of topological groups hold: where the restricted product has the restricted product topology, which is generated by restricted open rectangles of the form where is a