The Pontryagin duals of Q/Z and Q

Write S = {z ∈ C : |z| = 1}. A character of a locally compact abelian group G is a continuous group homomorphism G → S. We denote by Ĝ the set of characters of G, where for φ1, φ2 ∈ Ĝ and x ∈ G, we define (φ1φ2)(x) = φ1(x)φ2(x). We assign Ĝ the final topology for the family of functions {φ 7→ φ(x) : x ∈ G}, i.e., the coarsest topology on Ĝ so that for each x ∈ G, the function φ 7→ φ(x) is continuous Ĝ→ S. With this topology, it is a fact that Ĝ is itself a locally compact abelian group, called the Pontryagin dual of G. It is a fact that the Pontryagin dual of a discrete abelian group is compact and that the Pontryagin dual of compact abelian group is discrete. The Pontryagin duality theorem states that in the category of locally compact abelian groups, there is a natural isomorphism from the double dual functor to the identity functor. With the subspace topology inherited from R, one checks that a compact subset of Q has empty interior, and therefore Q is not locally compact. Thus, to work with the rational numbers in the category of locally compact abelian groups, we cannot use the subspace topology inherited from R. Rather, we assign Q the discrete topology. (Any abelian group is a locally compact abelian group when assigned the discrete topology.) From now on, when we speak about Q, unless we say otherwise it has the discrete topology. Because we use the discrete topology with Q, its Pontryagin dual Q̂ is a compact abelian group, which we wish to describe in a tractable way.