Exercise 8.3.5

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}}

Suppose that we have extensions $F \subset F_{i-1}\subset F_i \subset L$ such that $L$ is Galois over $F$ and $F_i$ is Galois over $F_{i-1}$. Prove that $|\Gal(F_i/F_{i-1})|$ divides $|\Gal(L/F)|$.

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}
$F \subset F_{i-1} \subset F_i \subset L$.

As $F\subset L$ is a Galois extension, $F_i \subset L$ and $F_{i-1} \subset L$ are also Galois, we have 
$$[L:F_i]  =\vert \Gal(L/F_i) \vert,\qquad  [L:F_{i-1}] =\vert \Gal(L/F_{i-1}\vert.$$

$\Gal(F_i/F_{i-1})\simeq \Gal(L/F_{i-1})/\Gal(L/F_i)$, thus $\vert \Gal(F_i/F_{i-1}) \vert$ divides $\vert \Gal(L/F_{i-1}\vert = [L/F_{i-1}]$.

As $ \vert \Gal(L/F)  \vert= [L:F] = [L:F_{i-1}][F_{i-1} : F]$, 
\begin{center}
$\vert \Gal(F_i/F_{i-1}) \vert$ divides $\vert \Gal(L/F)\vert $.
\end{center}
\end{proof}

Exercise 8.3.5

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Exercise 8.3.5

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