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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?

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    late_learner: I had mentioned the link in your 2nd paragraph as a comment to my answer below.2011-07-02

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The dual of the additive group is A_Q/Q. See http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/characterQ.pdf

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    @plusepsilon.de - the Pontryagin dual of $\Bbb Q^\times$ is a countable product of circle groups. So are you saying that $\Bbb A_f^\times / \Bbb Q^\times$ is a countable product of circle groups? If so, how?2015-11-10
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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).