Exercise 6.1.5

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}}

Prove the following inequalities:
\begin{enumerate}
\item[(a)]  $ | \Gal(\Q(\sqrt{2},\sqrt{3},\sqrt{5})/\Q)| \leq 8$
\item[(b)]$ | \Gal(\Q(\sqrt{p_1},\ldots,\sqrt{p_n})/\Q)| \leq 2^n$, where $p_1,\ldots,p_n$ are the first $n$ primes.
In each case, one can show that these are actually equalities.
\end{enumerate}

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}
\begin{enumerate}
\item[(a)]
As $\sqrt{2}$ is a root of $f_1=x^2-2$, $\sqrt{3}$ a root of $f_2=x^2-3$, and  $\sqrt{5}$ a root of $f_3=x^2-5$, Exercise 1 shows that
$$\vert \mathrm{Gal}(F(\sqrt{2},\sqrt{3},\sqrt{5})/F) \vert \leq \deg(f_1)\deg(f_2)\deg(f_3) = 8.$$

\item[(b)]
As $\sqrt{p_i}$ is a root of $f_i=x^2-p_i$, the same Exercise 1 shows that 
$$\vert \mathrm{Gal}(F(\sqrt{p_1},\ldots,\sqrt{p_n})/F) \vert \leq \deg(f_1)\cdots\deg(f_n) = 2^n.$$

\end{enumerate}
\end{proof}

Exercise 6.1.5

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Exercise 6.1.5

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