Let $p$ be an odd prime and $\alpha \in \mathbb{Z_p^*}$. Let $\mathcal{P} := \{ S \subset \mathbb{Z_p^*}: \lvert S \rvert = 2\}$, and define $\mathcal{C} := \{ \{\kappa, \lambda\} \in \mathcal{P} : \kappa\lambda = \alpha\}$. Note that for every $\kappa \in \mathbb{Z_p^*}$, there is a unique $\lambda \in \mathbb{Z_p^*}$ such that $\kappa \lambda = \alpha$, namely $\displaystyle \lambda := \frac{\alpha}{\kappa}$; moreover, $\kappa \ne \lambda$. Define $\mathcal{D} := \{ \{\kappa, \lambda\} \in \mathcal{P} : \kappa\lambda = 1\}$. Calculate the sets $\mathcal{C}$ and $\mathcal{D}$ in the case $p = 11$ and $\alpha = -1$.

$\mathcal{C}: \{\{1,10\}, \{2,5\}, \{3,7\}, \{4,8\}, \{6, 9\}\}$
$\mathcal{D}: \{\{2,6\}, \{3,4\}, \{5,9\}, \{7,8\}\}$

# posted by rot13(Unafba Pune) @ 10:13 PM