TITLE: What is the Pontryagin dual of the rationals?
QUESTION [14 upvotes]: 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?

REPLY [14 votes]: For those who don't know what is $A_Q$...  
Hewitt and Ross, Abstract Harmonic Analysis, p. 404.  The dual of the discrete rationals is described as an $\mathbf{a}$-adic solenoid.  An inverse limit of a sequence of circles, $T_n$, say, where the map of $T_{n+1}$ onto $T_n$ wraps around $n$ times.  
Their notes say this is due to Makoto Abe (1940) and independently to Anzai and Kakutani (1943).