Exercise 7.4.1

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}}

Give a detailed proof of Proposition 7.4.2:

Let $f\in F[x]$ be a monic irreducible separable cubic, where $F$ has characteristic $
e 2$. If $L$ is the splitting field of $f$ over $F$, then
$$
\Gal(L/F) \simeq 
\left\{
\begin{array}{ll}
 \Z/3\Z, &   \mathrm{if}\ \Delta(f)\ \mathrm{is}\  \mathrm{a}\  \mathrm{square}\  \mathrm{in} \ F,   \
  S_3,&        \mathrm{otherwise.}
\end{array}
ight.
$$

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}
Since $L$ is the splitting field of the separable polynomial $f$, $F \subset L$ is a Galois extension.

By Exercise 6.2.6, $f$ being irreducible and separable, $n = \vert \Gal(L/F) \vert $ is a multiple of $3 = \deg(f)$. Moreover $\Gal(L/F)$ is isomorphic to a subgroup $H$ of $S_3$, so $n \mid 6$: $n=3$ or $n=6$. Since $S_3$ has a unique subgroup of cardinality 3, namely $A_3 \simeq \Z/3\Z$,
$$\Gal(L/F) \simeq A_3 	ext{ or } \Gal(L/F) \simeq S_3.$$ 
By Theorem 7.4.1, since the characteristic of $F$ is different from 2, $\Gal(L/F) \simeq H \subset A_3$ if and only if $\sqrt{\Delta(f)} \in F$, therefore
$$
\Gal(L/F) \simeq 
\left\{
\begin{array}{ll}
 \Z/3\Z, &   \mathrm{if}\ \Delta(f)\ \mathrm{is}\  \mathrm{a}\  \mathrm{square}\  \mathrm{in} \ F,   \
  S_3,&        \mathrm{otherwise.}
\end{array}
ight.
$$
\end{proof}

Exercise 7.4.1

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Exercise 7.4.1

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