Exercise 9.32

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 $p$ be a prime, $p\equiv 1 \pmod 4$ and write $p = \pi \overline{\pi}, \pi \in \Z[i]$. Show $\chi_p(1+i) = i^{(p-1)/4}$.

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}

\begin{align*}
\chi_p(1+i) &= \chi_\pi(1+i) \chi_{\bar{\pi}}(1+i) \
&= \chi_\pi(1+i) \overline{\chi_\pi(1-i)} \qquad ( \mathrm{Prop.}\  \mathrm{9.8.3(c)})\
& =\frac{\chi_\pi(1+i)}{\chi_\pi(1-i) }= \chi_\pi(i) \qquad (\mathrm{since}\  (1-i)i = 1+i)\
&=i^{\frac{p-1}{4}}.
\end{align*}
The last equality is a consequence of the definition of $\chi_\pi$ : $\chi_\pi(i) \equiv i^{\frac{p-1}{4}} \pmod \pi$, and the classes of $1,i,i^2,i^3$ modulo $\pi$ are distinct.
\end{proof}

Exercise 9.32

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Exercise 9.32

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