Example 42.68.20. Let $R = \mathbf{Z}_ p$ and let $M = \mathbf{Z}_ p/(p^ l)$. Let $\varphi = p^ b u$ and $\varphi = p^ a v$ with $a, b \geq 0$, $a + b = l$ and $u, v \in \mathbf{Z}_ p^*$. Then a computation as in Example 42.68.19 shows that
\begin{eqnarray*} \det \nolimits _{\mathbf{F}_ p}(\mathbf{Z}_ p/(p^ l), p^ bu, p^ av) & = & (-1)^{ab}u^ a/v^ b \bmod p \\ & = & (-1)^{\text{ord}_ p(\alpha )\text{ord}_ p(\beta )} \frac{\alpha ^{\text{ord}_ p(\beta )}}{\beta ^{\text{ord}_ p(\alpha )}} \bmod p \end{eqnarray*}
with $\alpha = p^ bu, \beta = p^ av \in \mathbf{Z}_ p$. See Lemma 42.68.37 for a more general case (and a proof).
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