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I know that there is classification of local fields, but here is a closely related question: Can the additive group of $\mathbb{Q}$ be a proper dense subgroup of a locally compact abelian group, whose topology is complete, other than the p adic numbers or the reals? I think of this question more as a collection, and I guess I will have to try out various examples here.

1.Example by MattE: Consider $\alpha =(\alpha_1, \dots, \alpha_n) \in \mathbb{R}^n$ linearly independent over $\mathbb{Q}$, then the map $q \mapsto q \alpha:= ( q \alpha_1, \dots, q \alpha_n)$ becomes dense in the $n$ torus, i.e. $\mathbb{R}^n / \mathbb{Z}^n$, actually even more it becomes equidistributed in the following sense $\frac{1}{N}\sum\limits_{n \leq N} f(n \alpha) \rightarrow \int\limits_{\mathbb{R^N} / \mathbb{Z}^n} f( x) \mathrm{d} x.$

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    Okay that's what happens exactly for the finite adeles, that is $\mathbb{Q} \subset \mathbb{Q} \prod\limits_p \mathbb{Z}_p$, so my 2nd questions seems stupid in retrospective.2011-06-29

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It can surely be embedded densely in many such groups. E.g. it can be embedded into $(S^1)^n$ for any $n$. (Here $S^1$ is the circle group.)

To see this, choose an element $\alpha$ in $(S^1)^n$ whose powers are dense in $(S^1)^n$.

Now for inductively, for each integer $m$, choose $\alpha_m$ such that $\alpha_m^m = \alpha$, in a compatible way (i.e. so that if m' = d m, then $\alpha_{m'}^d = \alpha_m$). Then the $\alpha_m$ together generate a copy of $\mathbb Q$ inside $(S^1)^n$, which will be dense.

(A little more succintly, I am using the fact that $(S^1)^n$ is divisible, hence injective, to extend the embedding $\mathbb Z \hookrightarrow (S^1)^n$ to an embedding $\mathbb Q \hookrightarrow (S^1)^n$.)

Another way to think about this example, when $n = 2$ say, is that we take a line with irrational slope in $(S^1)^2$; this gives a dense copy of $\mathbb R$, which contains inside it a dense copy of $\mathbb Q$.

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    @Joel: Dear Joel, Yes, you're right! (I just bashed out the first thing that came to mind.) Thanks, and best wishes,2011-06-29