Exercise 13.2.20

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}}

In the Mathematical Notes to Section 10.3, we noted that the roots of the polynomial $x^5 - 4x^4 + 2x^3 + 4x^2 + 2x - 6 \in \Q[x]$ can be constructed using a marked ruler and compass. Show that this polynomial is not solvable by radicals over $\Q$.

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}
With the same procedures as in Exercise 19, we obtain
\begin{verbatim}
R.<x> = QQ[]
f = x^5 - 4*x^4 + 2*x^3 + 4 *x^2 + 2*x -6;f
\end{verbatim}
$$x^{5} - 4 x^{4} + 2 x^{3} + 4 x^{2} + 2 x - 6$$

\begin{verbatim}
f.is_irreducible()
\end{verbatim}
$$	ext{True}$$

\begin{verbatim}
theta = resolvent(f)[0]; theta.subs(Delta = f.discriminant()).expand()
\end{verbatim}
$$y^{6} - 360 \, y^{5} + 47856 \, y^{4} - 3025152 \, y^{3} + 103474944 \,
y^{2} - 1812875264 \, y + 14770999296
$$
\begin{verbatim}
res = rational_root(theta); res
\end{verbatim}
$$	ext{(False, None)}$$
\begin{verbatim}
f.discriminant().factor(),f.discriminant().is_square()
\end{verbatim}
$$(-1 \cdot 2^{4} \cdot 4003, 	ext{False})$$
So the Galois group of $f$ is $S_5$, and $f$ is not solvable by radicals over $\Q$.

Verification:
\begin{verbatim}
f.galois_group().gens()
\end{verbatim}
$$\langle (1,2), (1,2,3,4,5) angle
$$
\end{proof}

\subsection{Resolvents}

Exercise 13.2.20

Answers

0.1 Resolvents

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Exercise 13.2.20

Answers

0.1 Resolvents

Comments

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