Definition 27.1.1.
Throughout this section, view \(A = \ZZ_p\llbracket q-1 \rrbracket\) as a \(\delta\)-ring in which \(q\) is constant, and identify \(A/([p]_q)\) with \(\ZZ_p[\zeta_p]\) via the map taking \(q\) to \(\zeta_p\text{.}\) We will use frequently the fact that the ideals \((p,q-1)\) and \((p, [p]_q)\) of \(A\text{,}\) although distinct, do define the same topology on \(A\text{;}\) keep in mind that \((A, ([p]_q))\) is a prism but \((A, (q-1))\) is not.
We will also use on several occasions the congruence
\begin{equation}
\phi([p]_q) = q^{p(p-1)} + \cdots + q^p + 1 \equiv p \pmod{[p]_q}.\tag{27.1}
\end{equation}