Hello,

I need some clarification (maybe an example would help) on how to get the inverse of a p-adic integer.

The method used (in the paper) is for getting the inverse of $1-\alpha$. So, if you have a p-adic integer, say $\gamma$, which ends in $1$, you have to let $\gamma = 1 - \alpha$, and solve for $\alpha$, which will end with a $0$. Then the inverse of $\gamma = 1 + \alpha + \alpha^2 + ...$. Is this the correct way of getting the inverse of $\gamma$?

Now, if $\gamma$ ends in something else other than zero or one, what we are supposed to do is look for a digit $f$ so that $f\gamma$ will end in a one, and thus can be inverted by the above method (if it's correct ). Since we have the inverse of $f\gamma$, to get the inverse of $\gamma$, we multiply the inverse of $f\gamma$ by $f$ again. Is this correct?

2. If $\alpha$ ends in $d$, pick the unique $f$ such that $0 and $df\equiv1\bmod{p}$.

Now $\beta=f\alpha$ has a last digit of $1$. We then can find the inverse of $\beta$.

So $\beta\beta^{-1}=1\implies(\alpha)\cdot\left(f\beta^{-1}\right)=1\implies (\alpha)\cdot\left(f(f\alpha)^{-1}\right)=1\implies\alpha^{-1}=f(f\alpha)^{-1}$.