Representation of p-adic integers using primitive roots


MHF Helper
Nov 2010
Let \(\displaystyle p\) be an odd prime and let \(\displaystyle r\) be any positive integer that is a primitive root module \(\displaystyle p^2\). Let \(\displaystyle X_p = \varprojlim \Bbb{Z} / (p-1)p^n \Bbb{Z}\) be the inverse limit of the rings \(\displaystyle \Bbb{Z} / (p-1)p^n \Bbb{Z}\). This is useful because for any unit \(\displaystyle u \in \Bbb{Z}/p^n\Bbb{Z}\), there exists \(\displaystyle K \in \Bbb{Z} / (p-1)p^{n-1}\Bbb{Z}\) with \(\displaystyle r^k \equiv u \pmod{p^n}\) for all \(\displaystyle k \equiv K \pmod{(p-1)p^{n-1}}\). So, any element of the p-adic integers: \(\displaystyle x \in \Bbb{Z}_p\), you can represent it as \(\displaystyle x = p^a r^b\) for some \(\displaystyle a \in \mathbb{N}, b\in X_p\). I haven't seen this examined at all. Multiplication of terms in \(\displaystyle \Bbb{Z}_p\) would just correlate to addition of in \(\displaystyle X_p\). There is no operation in \(\displaystyle X_p\) that correctly models addition in \(\displaystyle \Bbb{Z}_p\). But, exponentiation in \(\displaystyle \Bbb{Z}_p\) closely correlates to multiplication in \(\displaystyle X_p\). Has anyone seen any examination of this type of representation for p-adic integers?