I know that the Pontryagin dual of $\mathbb{Q}/\mathbb{Z}$ is the group of profinite integers: $$\widehat{\mathbb{Z}}\overset{\textrm{def}}{=}\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}$$ under the addition operation, where $\mathbb{P}$ is the set of prime numbers ($2,3,5,7,\ldots$) and the $\mathbb{Z}_{p}$s are the rings of $p$-adic integers.
Now, let $\mathbb{F}$ be a purely real number field (ideally, $\overline{\mathbb{Q}}\cap\mathbb{R}$, the intersection of the field of algebraic numbers with $\mathbb{R}$). Then, what is the Pontryagin dual of $\mathbb{F}/\mathbb{Z}$?
My guess is that it would be:$$\widehat{\mathbb{F}/\mathbb{Z}}\cong\prod_{p\in\mathbb{P}_{\mathbb{F}}}\mathbb{Z}_{p}$$ where $\mathbb{P}_{\mathbb{F}}$ is the set of places/primes of $\mathbb{F}$, and $\mathbb{Z}_{p}$ is the ring of $p$-adic integers (i.e., the closure of $\mathbb{Z}\left[p\right]$ with respect to the $p$-adic absolute value). Is this correct? If not, what is $\widehat{\mathbb{F}/\mathbb{Z}}$?
Finally, insofar as Fourier analysis is concerned, am I correct in assuming that characters, fractional parts, Fourier transforms, haar measures and the like all work out to be formally equivalent to what one does when $p\in\mathbb{P}$? That is to say, assuming that I know how to do Fourier analysis on $\widehat{\mathbb{Q}/\mathbb{Z}}\cong\widehat{\mathbb{Z}}$, is there anything special I need to know about Fourier analysis on $\widehat{\mathbb{F}/\mathbb{Z}}$? Or do all the formulae generally work in the same way?