Exercise 9.45

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 $\pi = a + bi \in \Z[i]$ and $q\equiv 3 \ [4]$ a rational prime. Show $\pi^q \equiv \overline{\pi} \ [q]$.

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}
Let $\pi=a+bi \in \mathbb{Z}[i]$, and $q\equiv 3 \ [4]$ a rational prime.

As $\binom{q}{k} \equiv 0 \pmod q$ for $1\leq k \leq q-1$, the Fermat's Little Theorem gives
\begin{align*}
\pi^q &=(a+bi)^q\
&\equiv a^q+b^qi^q\ [q]\
&\equiv a + b i^3 \ [q]\
&= a-bi \
&= \overline{\pi}
\end{align*}

Conclusion : $\pi^q \equiv \bar{\pi}\ [q]$ ($\pi \in \mathbb{Z}[i]$, and $q\equiv 3 \ [4]$)
\end{proof}

Exercise 9.45

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Exercise 9.45

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