Exercise 2.1.47 (Revisited Wilson's theorem )

Question

If $r_1,r_2,\ldots,r_{p-1}$ is any reduced residue system modulo a prime $p$, prove that
$$\prod_{j=1}^{p-1} r_j \equiv -1 \pmod p.$$

Richard Ganaye · Accepted Answer

\begin{proof}

Since $\{r_1,r_2,\ldots,r_{p-1}\}$ and $\{1,2,\cdots,p-1\}$ are two reduced residue systems modulo $p$, there is a permutation $\sigma \in S_{p-1}$ such that
$$r_j \equiv \sigma(j) \pmod p,\qquad j=1,\ldots,p-1.$$
Thus
\begin{align*}
\prod_{j=1}^{p-1} r_j  &\equiv \prod_{j=1}^{p-1} \sigma(j) \pmod p
\end{align*}
By Wilson's theorem,
$$\prod_{j=1}^{p-1} \sigma(j) = \prod_{j=1}^{p-1} j = (p-1)! \equiv -1 \pmod p,$$
so 
$$\prod_{j=1}^{p-1} r_j \equiv -1 \pmod p.$$

\end{proof}

Exercise 2.1.47 (Revisited Wilson's theorem )

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Exercise 2.1.47 (Revisited Wilson's theorem )

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