Exercise 13.4.1

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}}

Verify the computations given in Example 13.4.1.
\begin{proof}
\be
The following Sage instructions:
\begin{verbatim}
R.<y,x1,x2,x3,sigma1,sigma2,sigma3,u1,u2,u3> = PolynomialRing(QQ, order = 'lex')
elt = SymmetricFunctions(QQ).e()
e = [elt([i]).expand(3).subs(x0=x1, x1=x2, x2=x3) for i in range(4)]
J = R.ideal(e[1]-sigma1, e[2]-sigma2, e[3]-sigma3)
G = J.groebner_basis()
S = y-(u1*x1 + u2*x2 + u3*x3)
S1 = S  * S.subs(x1=x1,x2=x3,x3=x2) * S.subs(x1=x2,x2=x1,x3=x3)
S1 = S1 * S.subs(x1=x2,x2=x3,x3=x1) * S.subs(x1=x3,x2=x1,x3=x2)
S = S1 * S.subs(x1=x3,x2=x2,x3=x1)
S = S.subs(u1=-1,u2=1,u3=2)
f = S.reduce(G).polynomial(y)
f
\end{verbatim}
give output $S(y)$:
\begin{align*}
& y^{6} - 4\sigma_{1}y^{5} + (2\sigma_{1}^{2} + 14\sigma_{2}) y^{4} + (8 \sigma_{1}^{3} - 44\sigma_{1} \sigma_{2} + 20\sigma_{3}) y^{3} \
& + (-7\sigma_{1}^{4} + 18 \sigma_{1}^{2} \sigma_{2} + 49 \sigma_{2}^{2} - 40\sigma_{1} \sigma_{3}) y^{2} + (-4 \sigma_{1}^{5} + 44\sigma_{1}^{3} \sigma_{2} \ & - 112\sigma_{1} \sigma_{2}^{2} - 20\sigma_{1}^{2} \sigma_{3} + 140\sigma_{2} \sigma_{3}) y+4\sigma_{1}^{6}  - 32 \sigma_{1}^{4}\sigma_{2} + 55 \sigma_{1}^{2} \sigma_{2}^{2} \
& + 36\sigma_{2}^{3} + 76 \sigma_{1}^{3} \sigma_{3} - 322  \sigma_{1} \sigma_{2} \sigma_{3} + 343 \sigma_{3}^{2}. 
\end{align*}
Using $\sigma_1 \mapsto -1,\sigma_2 \mapsto -2,\sigma_3 \mapsto 1$, we obtain:
\begin{verbatim}
f1 = f.subs(sigma_1=-1, sigma_2=-2, sigma_3=1); f1
\end{verbatim}
$$y^{6} + 4 y^{5} - 26 y^{4} - 76 y^{3} + 193 y^{2} + 240 y - 377$$
The Sage output coincides with Example 13.4.1:
\begin{verbatim}
factor(f1)
\end{verbatim}
$$(y^{3} + 2 y^{2} - 15 y - 29) \cdot (y^{3} + 2 y^{2} - 15 y + 13)$$
\ee\end{proof}

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}}

Exercise 13.4.1

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Exercise 13.4.1

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