Exercise 5.3.5

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 $f = x^7+x+1$, find all primes for which $f_p$ is not separable, and compute $\mathrm{gcd}(f_p,f'_p)$ as in (5.14).

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} The following Sage instructions give the wanted factorisation of the discriminant of $f$: \begin{verbatim} P. = PolynomialRing(QQ) f = x^7+x+1 g = diff(f(x),x) d = f.resultant(g);d \end{verbatim} \begin{center} 870199 \end{center} \begin{verbatim} f.discriminant() \end{verbatim} \begin{center} -870199 \end{center} \begin{verbatim} d.factor() \end{verbatim} \begin{center} $11 \cdot 239 \cdot 331$ \end{center} \begin{verbatim} P. = PolynomialRing(GF(11)) f = x^7 + x + 1; df = diff(f(x),x) gcd(f,df) \end{verbatim} \begin{center} $x+3$ \end{center} \begin{verbatim} factor(f) \end{verbatim} \begin{center} $(x + 3)^{2} \cdot (x^{5} + 5 x^{4} + 5 x^{3} + 2 x^{2} + 9 x + 5)$ \end{center} \begin{verbatim} P. = PolynomialRing(GF(239)) f = x^7 + x + 1; df = diff(f(x),x) gcd(f,df) \end{verbatim} \begin{center} $x+41$ \end{center} \begin{verbatim} factor(f) \end{verbatim} \begin{center} $(x + 41)^{2} \cdot (x^{5} + 157 x^{4} + 24 x^{3} + 122 x^{2} + 81 x + 30)$ \end{center} \begin{verbatim} P. = PolynomialRing(GF(331)) f = x^7 + x + 1; df = diff(f(x),x) gcd(f,df) \end{verbatim} \begin{center} $x+277$ \end{center} \begin{verbatim} factor(f) \end{verbatim} \begin{center} $(x + 277)^{2} \cdot (x^{2} + 188 x + 203) \cdot (x^{3} + 251 x^{2} + 84x + 80).$ \end{center} \bigskip So $$ \mathrm{gcd}(f_p,f'_p) = \left\{ \begin{array}{ll} x+3, & p = 11, \ x + 41, & p = 239, \ x + 277, & p = 331, \ 1 & \mathrm{otherwise.} \end{array} ight. $$ \end{proof}

Exercise 5.3.5

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Exercise 5.3.5

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