finite subset of the set of all places and are open sets. Proof. Prove the identity for ; the other two follow similarly. First show the two sets are equal: In going from line 2 to 3, as well as have to be in meaning for almost all and for almost all Therefore, for almost all Now, it is possible to show the topology on the left-hand side equals the topology on the right-hand side. Obviously, every open restricted rectangle is open in the topology of the idele group. On the other hand, for a given which is open in the topology of the idele group, meaning is open, so for each there exists an open restricted rectangle, which is a subset of and contains Therefore, is the union of all these restricted open rectangles and therefore is open in the restricted product topology. Lemma. For each set of places, is a locally compact topological group. Proof. The local compactness follows from the description of as a restricted product. It being a topological group follows from the above discussion on the group of units of a topological ring. A neighbourhood system of is a neighbourhood system of Alternatively, take all sets of the form: where is a neighbourhood of and for almost all Since the idele group is a locally compact, there exists a Haar measure on it. This can be normalised, so that This is the normalisation used for the finite places. In this equation, is the finite idele group, meaning the group of units of the finite adele ring. For the infinite places, use the multiplicative lebesgue measure The idele group of a finite extension Lemma. Let be a finite extension. Then: where the restricted product is with respect to Lemma. There is a canonical embedding of in Proof. Map to with the property for Therefore, can be seen as a subgroup of An element is in this subgroup if and only if his components satisfy the following properties: for and for and for the same place of The case of vector spaces and algebras The idele group of an algebra Let be a finite-dimensional algebra over Since is not a topological group with the subset-topology in general, equip with the topology similar to above and call the idele group. The elements of the idele group are called idele of Proposition. Let be a finite subset of containing