I am reading this paper (the contents aren't too important, but it might add some context). I am confused on a step that occurs in Lemma 4 (page 391).
Namely, we are working modulo an even prime power, so we are working modulo $q=p^{2\beta}$. The author is looking at units modulo $q$ and states
Set $x=u+vp^\beta$ where $u,v$ arnge over the residue classes $\pmod{p^\beta}$, $(u,q)=1$. We find that $\overline{x}\equiv \overline{u}-\overline{u}^2vp^{\beta +1}\pmod{p^{2\beta+1}}$
I will remark that $\overline{u}$ is the inverse of $u$ (in what I am assuming is $\pmod{p^\beta})$. I am just confused as to how this was derived or if this is even correct. Since when I multiply things out (I will say that $u\overline{u}=1+kp^\beta$ for some $k\in\mathbb{Z}$. Then I end up getting that \begin{align*} x\overline{x} & \equiv (u+vp^\beta)(\overline{u}-\overline{u}^2vp^{\beta+1})\pmod{p^{2\beta+1}}\\ &\equiv u\overline{u}+\overline{u}vp^{\beta}-u\overline{u}^2vp^{\beta+1}\pmod{p^{2\beta+1}}\\ &\equiv (1+kp^\beta)+\overline{u}vp^\beta-(1+kp^\beta)\overline{u}vp^{\beta+1}\pmod{p^{2\beta+1}}\\ &\equiv 1+p^\beta\left(k+\overline{u}v-\overline{u}vp\right)\pmod{p^{2\beta+1}} \end{align*}
It isn't clear to me that the resulting thing in the end should become $1\pmod{p^{2\beta+1}}$. I think this is just wrong as written and might be a typo, but if it is correct then it eludes me. If it is a typo I think it should be corrected to $\overline{x}\equiv \overline{u}-\overline{u}^2vp^{\beta}$, then we do get some nicer cancelation, but we would then have to work modulo $p^{2\beta}$ which I think should be fine since that is the modulus we are working with. I think this would make more sense as we are working modulo $p^{2\beta}$ rather than $p^{2\beta+1}$ (which is the subsequent case).
For a bit more of a general context, the author is working with Salie sums, and they are proving a formula for Salie sums whose moduli are prime powers. In the case I am looking at the sum is over all units modulo $p^{2\beta}$ they are writing an arbitrary unit $x=u+vp^\beta$ as described above, so in the Salie-sum we need only what $\overline{x}$ is modulo $p^{2\beta}$ which leads me to believe it is a typo. I will note that if I assume it is a typo and that $\overline{x}\equiv\overline{u}-\overline{u}^2vp^\beta\pmod{p^{2\beta}}$, then my calculations then give that $$ x\overline{x}\equiv 1+kp^\beta\pmod{p^{2\beta}} $$ which still won't give it to me (unless I knew that $k\equiv 0\pmod{p^\beta}$ which I don't see). Although this seems more promising, I think that it is probably something easy that I am just not seeing.
