Exercise 13.4.5

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}}

Use the method of part (a) of Exercise 4 to derive the factorization of $s_u(y)$ given in Example 13.4.4.
\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.reduce(G).polynomial(y)
sf = s.subs(sigma1=0,sigma2=0,sigma3=1)
dec = sf.factor(); dec 
\end{verbatim}
give the same output as in Example 13.4.4:
\begin{align*}
s_u(y)=&       	
(y^{2} + \left(-2 u_{1} + u_{2} + u_{3}ight) y + u_{1}^{2} -  u_{1}
u_{2} + u_{2}^{2} -  u_{1} u_{3} -  u_{2} u_{3} + u_{3}^{2}) \cdot\
& 
(y^{2} + \left(u_{1} - 2 u_{2} + u_{3}ight) y + u_{1}^{2} -  u_{1}
u_{2} + u_{2}^{2} -  u_{1} u_{3} -  u_{2} u_{3} + u_{3}^{2}) \cdot\
&
(y^{2} + \left(u_{1} + u_{2} - 2 u_{3}ight) y + u_{1}^{2} -  u_{1}
u_{2} + u_{2}^{2} -  u_{1} u_{3} -  u_{2} u_{3} + u_{3}^{2})
\end{align*}

\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 13.4.5

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Exercise 13.4.5

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