n-copies of integers as an algebraic subgroup of p-adic integers

Question

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?

Answer 1

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 $\phi(a)=\sum_{i=0}^{n-1}a_iq^{i/n}$.