Exercise 4.2.4

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}}

For each of the following polynomials, use a computer to determine whether it is irreducible over the given field.
\be
\item[(a)] $x^4 + x^3 + x^2 + x + 2$ over $\Q$.
\item[(b)] $3x^6 + 6 x^5 + 9x^4 + 2x^3 + 3x^2 + 1$ over $\Q$ and $\Q(\sqrt[3]{2})$.
\ee

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}
\begin{enumerate}
\item[(a)]
With Sage, the instructions
\begin{verbatim}
factor(x^4+x^3+x^2+x+2)
factor(3*x^6+6*x^5+9*x^4+2*x^3+3*x^2+1);
\end{verbatim}
give the same polynomials.

So $x^4+x^3+x^2+x+2$ and $3x^6+6x^5+9x^4+2x^3+3x^2+1$ are irreducible over $\Q$.

\item[(b)]
The instructions

\begin{verbatim}
K = NumberField(x^3-2, 'a'); L.<X> = PolynomialRing(K)
p =  3*x^6 + 6*x^5 + 9*x^4 + 2*x^3 + 3*x^2 + 1
u = factor(p)
\end{verbatim}
give the following decomposition, where $a=\sqrt[3]{2}$:

$3x^6+6x^5+9x^4+2x^3+3x^2+1 =$

$\frac{1}{3} (3x^2 + (-a^2 + a + 2)x + a^2 - a + 1)	imes$

$(3x^4 + (a^2 - a + 4)x^3 + (a + 4)x^2 + (-a^2 - a)x + a + 1)$.

Thus $3x^6+6x^5+9x^4+2x^3+3x^2+1$ is not irreducible over $\Q(\sqrt[3]{2})$.
\end{enumerate}
\end{proof}

Exercise 4.2.4

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Exercise 4.2.4

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