Exercise 2.3.2

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 Example 2.3.1 or 2.3.2 to find the cubic polynomial whose roots are the fourth powers of the roots of the polynomial $y^3+2y^2-3y+5$.

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}
Same method in Sage as in Ex.2.3.1
\begin{verbatim}
e = SymmetricFunctions(QQ).e()
e1, e2, e3 = e([1]).expand(3),e([2]).expand(3),e([3]).expand(3)
R.<x0,x1,x2,y1,y2,y3> = PolynomialRing(QQ, order = 'degrevlex')
J = R.ideal(e1-y1, e2-y2, e3-y3)
G = J.groebner_basis()
f = (x-x0^4) * (x-x1^4) * (x-x2^4)
coeffs = f.coefficients(x, sparse = False)
coeffs = map(lambda c : R(c), coeffs)
coeffs
\end{verbatim}
$$\left[- x_{0}^{4} x_{1}^{4} x_{2}^{4}, x_{0}^{4} x_{1}^{4} + x_{0}^{4}
x_{2}^{4} + x_{1}^{4} x_{2}^{4}, - x_{0}^{4} -  x_{1}^{4} -  x_{2}^{4},
1ight]
$$
\begin{verbatim}
var('sigma_1,sigma_2,sigma_3,y')
ncoeffs = [c.reduce(G) for c in coeffs]
nncoeffs = [c.subs(y1 = sigma_1,y2 = sigma_2,y3 = sigma_3) for c in ncoeffs]
nncoeffs
\end{verbatim}
$$\left[- x_{0}^{4} x_{1}^{4} x_{2}^{4}, x_{0}^{4} x_{1}^{4} + x_{0}^{4}
x_{2}^{4} + x_{1}^{4} x_{2}^{4}, - x_{0}^{4} -  x_{1}^{4} -  x_{2}^{4},
1ight]
$$
\begin{verbatim}
nnncoeffs = [c.subs(sigma_1 = -2, sigma_2 = -3, sigma_3 = -5) for c in nncoeffs]
p = sum(nnncoeffs[i]*y^i for i in range(1+f.degree(x)))
p
\end{verbatim}
$$y^{3} - 122 \, y^{2} - 379 \, y - 625.$$

So the cubic polynomial whose roots are the fourth powers of the roots of the polynomial $y^3+2y^2-3y+5$ is
$$y^{3} - 122 \, y^{2} - 379 \, y - 625.$$
\end{proof}

Exercise 2.3.2

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Exercise 2.3.2

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