Exercise 9.1.15

Question

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ewcommand{\ssi} {si et seulement si }

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ewcommand{\C}{\mathbb{C}}

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Let $\mu$ be the M"obius function defined in Exercise 14. Prove that
$$\Phi_n(x) = \prod_{d\mid n} (x^d-1)^{\mu(n/d)}.$$

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}
Our starting point is $$F(n) = x^n - 1 = \prod\limits_{ d \mid n} \Phi_d \hspace{1cm} (n \geq 1).$$
It is sufficient to copy the proof of the M"obius Inversion Formula in multiplicative notations:
\begin{align*}
   \prod_{d \mid n} (x^d - 1)^{\mu \left (\frac{n}{d} ight )} &=  \prod\limits_{e \vert n}  (x^{\frac{n}{e}} -1)^{\mu(e)}\
           &= \prod\limits_{e \vert n} \ \prod\limits_ {d \vert \frac{n}{e}} \Phi_d^{ \mu(e) } \
            &=\prod\limits_{d \vert n}\  \prod_{e \vert \frac{n}{d}}\Phi_d^{\mu(e)}
            			\hspace{1cm} (\mathrm{ since }\  e\vert n \  \mathrm{and}   \ d \vert \frac{n}{e}  \iff d \vert n\  \mathrm{and}  \ e \vert \frac{n}{d}) \
           & = \prod\limits_{d \vert n}   \Phi_d^{\sum\limits_{\ e \vert \frac{n}{d}}\mu(e)} \
           &= \Phi_n,
 \end{align*}         
           since by Exercise 14, $\sum_{\ e \vert \frac{n}{d}} \mu(e) 
ot = 0$ only if $\frac{n}{d}=1$, that is $d =n$, so the product is $\Phi_n$.

Conclusion :            
$$\Phi_n(x) = \prod\limits_{d \mid n} (x^d - 1)^{\mu \left (\frac{n}{d} ight )}\quad (n\geq 1).$$
\end{proof}

Exercise 9.1.15

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Exercise 9.1.15

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