Exercise 8.6.3

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}}

Show that the polynomial $f = x^4 -4x^2 +x + 1$ of Example 8.6.7 is irreducible over $\Q$ and has four real roots.

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} By Gauss Lemma (Theorem A.3.2), it is sufficient to prove that $f$ has no non trivial factorization in $\Z[x]$. Since the reduction modulo 2 of $f$ is $\overline{f} = x^4 +x+1$ is of the same degree, it is sufficient to prove that $\overline{f}$ is irreducible in $\F_2[x]$ (a non trivial factorization in $\Z[x]$ would give a factorisation in $\F_2[x]$ by projection). This is the case if $\overline{f}$ has no root in $\F_4 \supset \F_2$ (an irreducible factor $\overline{g}$ of $\overline{f}$ of degree 2 would give a root of $g$ in $\F_2[x]/\langle g angle \simeq \F_4$). As any element $\alpha \in \F_4$ satisfies $\alpha^4 = \alpha = -\alpha$, $\overline{f}(\alpha) = \alpha^4+\alpha +1 = 1 e 0$, so $\overline{f}$ is irreducible over $\F_2$. Therefore \begin{center} $x^4 -4x^2 +x + 1$ is irreducible over $\Q$ \end{center} \bigskip $f(-3) = 43>0, f(-2) = -1<0,f(0) = 1>0, f(1) = -1<0, f(2) = 3>0$. As $f$ is continuous, the Intermediate Value Theorem shows the existence of the roots $x_1 \in ]-3,-2[, x_2 \in ]-2,0[, x_3 \in ]0,1[, x_4 \in ]1,3[$. As $\deg(f) = 4$, $f$ has no other root, so all the roots of $f$ are real. \end{proof}

Exercise 8.6.3

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Exercise 8.6.3

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