What is the Pontryagin dual of the rationals? Endow the rational numbers (or any global field) with the discrete topology, what will be the (compact) Pontryagin dual of the additive group and of the multiplicative group? I am suprised nobody mentioned this: but the part of the question of the additive group of the rational is answered here already: Representation theory of the additive group of the rationals?

The dual of the additive group $\mathbf Q$ with the discrete topology is $A_\mathbf Q/\mathbf Q$, where $A_\mathbf Q$ is the adele ring of $\mathbf Q$ (viewed as an additive group). See a proof here.

The standard topology on $A_\mathbf Q$ makes it locally compact and $\mathbf Q$ (embedded diagonally) is a discrete subgroup for which the quotient topology on $A_\mathbf Q/\mathbf Q$ is compact, as it needs to be if it's going to be the Pontryagin dual of a discrete abelian group.