Exercise 5.3.15

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}}

Let $f$ be the polynomial considered in Example 5.3.9. Use Maple or Mathematica to factor $f$ and to verify that the product of the distinct irreducible factors of $f$ is the polynomial given in (5.10).

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}
 Sage instructions:

\begin{verbatim}
f = x^11-x^10+2*x^8-4*x^7+3*x^5-3*x^4+x^3+3*x^2-x-1; f
\end{verbatim}
\begin{center}     	
$x^{11} - x^{10} + 2 \, x^{8} - 4 \, x^{7} + 3 \, x^{5} - 3 \, x^{4} + x^{3} + 3 \, x^{2} - x - 1$
\end{center}

\begin{verbatim}
f1 = f.derivative(); f1
\end{verbatim}
\begin{center}     	
$11 \, x^{10} - 10 \, x^{9} + 16 \, x^{7} - 28 \, x^{6} + 15 \, x^{4} -12 \, x^{3} + 3 \, x^{2} + 6 \, x - 1$
\end{center}

\begin{verbatim}
d = gcd(f,f1); d
\end{verbatim}
\begin{center}     	
$x^{6} - x^{5} + x^{3} - 2 \, x^{2} + 1$
\end{center}

\begin{verbatim}
p = (f/d).simplify_rational(); p
\end{verbatim}
\begin{center}     	
$x^{5} + x^{2} - x - 1$
\end{center}

\begin{verbatim}
v = p.factor(); v
\end{verbatim}
\begin{center}     	
${\left(x^{3} + x + 1ight)} {\left(x + 1ight)} {\left(x - 1ight)}$
\end{center}

\begin{verbatim}
w = d.factor(); w
\end{verbatim}
\begin{center}     	
${\left(x^{3} + x + 1ight)} {\left(x + 1ight)} {\left(x -
1ight)}^{2}$
\end{center}

\begin{verbatim}
s = f.factor(); s
\end{verbatim}
\begin{center}     	
${\left(x^{3} + x + 1ight)}^{2} {\left(x + 1ight)}^{2} {\left(x -1ight)}^{3}$
\end{center}
\begin{verbatim}
s.expand()
\end{verbatim}
\begin{center} 
$x^{11} - x^{10} + 2 \, x^{8} - 4 \, x^{7} + 3 \, x^{5} - 3 \, x^{4} + x^{3} + 3 \, x^{2} - x - 1$
\end{center}

\end{proof}

Exercise 5.3.15

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Exercise 5.3.15

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