Exercise 8.20

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}}

Generalize Proposition 8.6.1 by finding an explicit formula for the number of solutions to $a_1x_1^2+a_2x_2^2+\cdots+a_rx_r^2 = 1$ in $\F_p$.

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}
Write $\chi$ the Legendre character.
\begin{align*}
N(a_1x_1^2+\cdots+a_r x_r^2=1)&=\sum\limits_{a_1u_1+\cdots+a_r u_r=1}N(x_1^2=u_1)\cdots N(x_r^2=u_r)\
&=\sum\limits_{a_1u_1+\cdots+a_r u_r=1}(1+\chi(u_1))\cdots(1+\chi(u_r))  \hspace{0.5cm}(v_i = a_i u_i)\
&=\sum\limits_{v_1+\cdots+v_r=1}(1+\chi(a_1)^{-1}\chi(v_1))\cdots(1+\chi(a_r^{-1})\chi(v_r))\
&=p^{r-1} + \chi(a_1^{-1})\cdots\chi(a_r^{-1})J(\chi,\chi,\cdots,\chi).
\end{align*}

$\chi(a_i^{-1}) = \overline{\chi(a_i)} = \chi(a_i) = \legendre{a_i}{p}.$

$J(\chi,\chi,\cdots,\chi)$ is computed in Chapter 8 Section 6. We obtain
$$
\left\{
\begin{array}{ccll}
 N(a_1x_1^2+\cdots+a_r x_r^2=1)  & =  & p^{r-1}+\legendre{a_1}{p}\cdots\legendre{a_r}{p}(-1)^{\frac{r-1}{2}\frac{p-1}{2}}p^{\frac{r-1}{2}}&\mathrm{if}\ r\ \mathrm{is}\ \mathrm{odd}, \
  &  = &p^{r-1}-\legendre{a_1}{p}\cdots\legendre{a_r}{p}(-1)^{\frac{r}{2}\frac{p-1}{2}}p^{\frac{r}{2}-1}&\mathrm{if}\ r\  \mathrm{is}\ \mathrm{even}.
\end{array}
ight.
$$
\end{proof}

Exercise 8.20

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Exercise 8.20

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