Exercise 11.2.3

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

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{\U}{\mathbb{U}}

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

ewcommand{\im}{\,\mathrm{Im}\,}

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

ewcommand{\Gal}{\mathrm{Gal}}

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

Use Theorem 11.2.4 to compute $N_6$ and $N_{36}$.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

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{\U}{\mathbb{U}}

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

ewcommand{\im}{\,\mathrm{Im}\,}

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

ewcommand{\Gal}{\mathrm{Gal}}

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

\begin{proof}
 By Theorem 11.2.4,
 $$
 N_n = \frac{1}{n} \sum_{m\mid n} \mu(m) p^{\frac{n}{m}}.
 $$
\begin{align*}
N_6 &= \frac{1}{6}(\mu(1)p^6+\mu(2)p^3+\mu(3)p^2+\mu(6)p)\
&=\frac{1}{6}(p^6-p^3-p^2+p)
\end{align*}

The $9 = 3	imes 3$ factors $d$ of $36=2^2	imes 3^2$, and the corresponding values of $\mu(d)$ are
$$
\begin{array}{c|c|c|c|c|c|c|c|c|c}
d         &1 & 2  & 3   &4  &6  & 9 & 12 & 18 & 36   \
\hline
\mu(d) & 1 & -1 & -1 & 0 & 1 & 0  & 0   & 0  & 0  
\end{array}
$$

\begin{align*}
N_{36} &= \frac{1}{36}(p^{36}-p^{18} -p^{12}+p^6)
\end{align*}
\end{proof}

Exercise 11.2.3

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Exercise 11.2.3

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