Question about the multiplicative inverse of a p-adic number

Question

I am reading the text of Gouvea about p-adic numbers and I am trying to show that every $\alpha\in \mathbb{Q}_{p}$ has a multiplicative inverse. In the case where $$\alpha =\sum_{i\geq 0}a_{i}p^{i}, \ \text{where} \ a_{0}\neq 0$$ I think I understood the logic, but the other case is a problem for me. For example, if I would like to find the inverse of the 5-adic number $$\alpha =3\cdot 5 + 4\cdot 25$$ what would be the strategy? Because I really don't see how I can find a $5$-adic number which can give me $1$... Can someone help me? Thanks before.

Answer 1

The inverse will not be a $5$-adic integer. So its expansion will have one or more negaive powers of $5$.

Your number is $5$ times $3+4\cdot 5$. If you invert $3+4\cdot 5$, and multiply that result by $5^{-1}$, you get the inverse of $3\cdot 5 + 4\cdot 5^2$.