TITLE: Finding quotient of $7/5^3=Exp(2\pi i 7/5^3)\in Z(5^{\infty})$ by $20$ or 20-th root
QUESTION [0 upvotes]: Find quotient of $7/5^3=Exp(2\pi i 7/5^3)\in Z(5^{\infty})$ by $20$ or in other words, find 20-th root of $7/5^3$ in $Z(5^{\infty})$. 
The troublesome part is $(20,5)=5$. So I cannot use coprimeness of $7$ and $20$ to generate 1 naively. How should I find such an element systematically?

REPLY [2 votes]: Note the Prufer $5$-group $\mathbb{Z}(5^\infty)=\mathbb{Z}[1/5]/\mathbb{Z}$ is a module over the $5$-adic integers $\mathbb{Z}_5$. Calculating a fourth root amounts to multiplying by the $5$-adic number $1/4$, which can be expanded by a geometric series using the fact that $4=5-1$. This is equivalent to computing $4^{-1}$ mod sufficiently high powers of $5$ to subsquently multiply against an element of $\mathbb{Z}(5^\infty)$.
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