Exercise 5.10

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

Let $r_1, r_2, \ldots, r_{(p-1)/2}$ be the quadratic residues between
$1$ and $p$. Show that their product is congruent to $1 \pmod p$ if $p
\equiv 3 \pmod 4$, and to $-1$ if $p \equiv 1 \pmod 4$.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
We have proved in Ex. 5.9 that
$$\left[\left(\frac{p-1}{2}ight)!ight]^2 \equiv (-1)^{(p+1)/2} \pmod p.$$
The application 
$ f:
\left\{
\begin{array}{ccc}
\{\overline{1},\overline{2},\ldots,\overline{(p-1)/2}\}  & \mapsto    & \{ \overline{r_1},\overline{ r_2}, \ldots, \overline{r_{(p-1)/2}}\} \
  x &  \mapsto  &  x^2   
\end{array}
ight.
$ is a bijection, so
$$\prod_{i=1}^{(p-1)/2} r_i \equiv \left[\left(\frac{p-1}{2}ight)!ight]^2 \pmod p,$$
so
$$\prod_{i=1}^{(p-1)/2} r_i \equiv (-1)^{(p+1)/2} \pmod p.$$
That is to say, the product of the quadratic residues between $1$ and $p$ is congruent to $1 \pmod p$ if $p
\equiv 3 \pmod 4$, and to $-1$ if $p \equiv 1 \pmod 4$.
\end{proof}

Exercise 5.10

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Exercise 5.10

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