Exercise 10.3.9

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}}

Prove Corollary 10.3.9.

\bigskip

{\bf Corollary 10.3.9} Let $f(x) \in \Q[x]$ be a polynomial of degree $\leq 4$. Then the roots of $f(x)$ are origami numbers, i.e, we can solve $f(x) = 0$ by origami.

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} Let $\alpha$ be a root of $f(x)$, with $d = \deg(f) \leq 4$.

Let $L$ be the splitting field of $\alpha$.

Recall that the Galois group $\Gal(L/\Q)$ is isomorphic to a subgroup of $S_4$, so 
$$ |\Gal(L/\Q)| = [L:\Q]\ \mathrm{divides} \  4!.$$
So $[L:\Q]$ divides $4! = 2^3 	imes 3$ , therefore $[L:\Q] = 2^a 3^b$ for some $a,b$ with  $0\leq a \leq 3,\  0 \leq b \leq 1$. By Theorem 10.3.6 and Exercise 8, $\alpha$ is an origami number.

\end{proof}

Exercise 10.3.9

Answers

Comments

Exercise 10.3.9

Answers

Comments

Add answer