Exercise 1.2.3

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}}

Prove (1.20)

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}
By (1.17), $\Delta = -4p^3-27q^2$.

Using (1.19), we obtain

\begin{align*}
p^3 &= \left( - \frac{b^2}{3}+cight)^3\
&=-\frac{b^6}{27} +  \frac{1}{3} b^4c - b^2c^2 + c^3,\
-4p^3 &= \frac{4}{27}b^6 - \frac{4}{3} b^4c +4 b^2c^2 -4 c^3,\
\end{align*}
\begin{align*}
q^2 &= \left(\frac{2b^3}{27} - \frac{bc}{3} + d ight)^2\
&=\frac{4}{27^2} b^6 +\frac{b^2c^2}{9} + d^2 -\frac{4b^4c}{3	imes 27} + \frac{4b^3d}{27} - \frac{2bcd}{3}\
-27q^2&= -\frac{4}{27} b^6 - 3 b^2c^2 - 27 d^2+ \frac{4}{3} b^4c - 4 b^3d + 18bcd.\
\end{align*}
So
\begin{align*}
\Delta&= -4p^3-27q^2\
&= b^2c^2 + 18bcd-4c^3-4b^3d-27d^2.
\end{align*}
\end{proof}

Exercise 1.2.3

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Exercise 1.2.3

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