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Let $\mathbb{Z}_p$ denote the additive group of $p$-adic integers. The group of integers $\mathbb{Z}$ can be viewed as a dense subgroup of $\mathbb{Z}_p$ generated by the element that projects to $1$ mod $p^n$ for all $n$. The same should be possible for the group $\mathbb{Z}^n$, but I don't see how to describe the homomorphism $\mathbb{Z}^n\hookrightarrow\mathbb{Z}_p$ on generators even for the case $n=2$. Any hints?

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  • $\begingroup$ (1) Show that $\mathbf{Q}_p$ does not have finite dimension over $\mathbf{Q}$. (2) Deduce that there exists a $\mathbf{Q}$-free family $a_1,\dots,a_n$ in $\mathbf{Q}_p$ (3) Deduce that there exists a $\mathbf{Q}$-free family $b_1,\dots,b_n$ in $\mathbf{Z}_p$. (4) Conclude. [Alternative to (3) is to find explicit families: either use algebraic numbers as in Neil's answer, or use a transcendental element.] $\endgroup$ – YCor Oct 4 '17 at 11:52
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Choose a prime $q$ congruent to $1$ mod $p$, so $q^{1/n}$ exists in $\mathbb{Z}_p$. Then define $\phi\colon\mathbb{Z}^n\to\mathbb{Z}_p$ by $\phi(a)=\sum_{i=0}^{n-1}a_iq^{i/n}$.

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