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

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

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

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

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