Exercise 12.3.7

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}}

Let $f \in F[x]$ be monic of degree $n>0$ with discriminant $\Delta(f) \in F$. Use Exercise 6 to show that thre are $A,B \in F[x]$ with $\deg(A) \leq n-2, \deg(B) \leq n- 1$, such that the coefficients of $A$ and $B$ are polynomials in the coefficients of f and $Af + B f' = \Delta(f)$.

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}
Set $f = x^n - c_1 x^{n-1} + \cdots +(-1)^n c_0$ any monic polynomial of degree $n$.

By Exercise 6, there exist $	ilde{A}, 	ilde{B} \in F[\sigma_1,\ldots,\sigma_n,x]$ such that
$$	ilde{A} 	ilde{f} + 	ilde{B} 	ilde{f}' = \Delta,\ \deg(	ilde{A}) \leq n-2, \deg(	ilde{B}) \leq n-1.$$
The evaluation $\sigma_i 	o c_i$ maps $\Delta$ to $\Delta(f)$, $	ilde{f}$ to $f$, $	ilde{f'}$ on $f'$. Write $A(x) = 	ilde{A}(c_1,\cdots,c_n,x), B(x) =  	ilde{B}(c_1,\ldots,c_n,x)$, so the evaluation maps $	ilde{A}, 	ilde{B}$ to $A,B$, and $\deg(A) \leq \deg(	ilde{A}), \deg(B) \leq 	ilde{B}$. Since $\Delta(f) = \Delta(c_1,\ldots,c_n)$ by 2.30, the evaluation of the two members of  $	ilde{A} 	ilde{f} + 	ilde{B} 	ilde{f}' = \Delta$ gives
$$Af + B f' = \Delta(f),\ \deg(A) \leq n-2, \deg(B) \leq n-1.$$

\end{proof}

Exercise 12.3.7

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Exercise 12.3.7

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