Exercise 6.11

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}}

By evaluating $\sum_t (1+(t/p)) \zeta^t$ in two ways, prove that $g = \sum_t \zeta^{t^2}$.

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}
For $ a \in \F_p$, Write $N[x^2 = a]$ the number of solutions of the equation $x^2 = a$ in $\F_p$.
We know from Ex. 5.2 that $N[x^2 = a] = 1 + (a/p)$. Therefore
\begin{align*}
\sum_{t=0}^{p-1} \zeta^{t^2} &= \sum_{\overline{t} \in \F_p} \zeta^{t^2} \
 & = \sum_{\overline{a} \in \F_p} N[x^2 = a] \zeta^a\
&=\sum_{\overline{t} \in \F_p} \left(1 + \legendre{t}{p} ight) \zeta^t\
&=\sum_{\overline{t} \in \F_p} \zeta ^t + \sum_{\overline{t} \in \F_p}  \legendre{t}{p} \zeta^t\
 &=\sum_{t = 0} ^{p-1}  \legendre{t}{p} \zeta^t\
 &= g
\end{align*}
\end{proof}

Exercise 6.11

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Exercise 6.11

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