Exercise 8.6.2

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 $F\subset K$ be a real radical extension and suppose that $F \subset M \subset \R$. Prove that $M\subset MK$ is a real radical extension.

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}
As the extension $F\subset K$ is radical, there exist  $\gamma_1, \ldots,\gamma_n \in K\subset \R$ such that the subfields  $K_i = F(\gamma_1,\ldots,\gamma_i),\ i=0,\ldots,n$ of $K$ satisfy
$$K_0 = F \subset K_1 \subset \cdots \subset K_n = K,$$
and $K_i = K_{i-1}(\gamma_i), \gamma_i \in K_i, \gamma_i^{m_i} \in K_{i-1}, m_i>0$.

Let $M_0 = M, M_i = M(\gamma_1,\ldots, \gamma_i)$. Then $K_0=F  \subset M_0 = M$ and $K_i \subset M_i$. Moreover
$$M_0 = M \subset M_1\subset\cdots\subset M_n = M(\gamma_1,\ldots,\gamma_n),$$
and $M_i = M_{i-1}(\gamma_i), \gamma_i^{m_i} \in K_{i-1} \subset M_{i-1}$, so 
\begin{center}
$M \subset M(\gamma_1,\ldots,\gamma_n) = M_n$ is a real radical extension.
\end{center}
Moreover, as $F \subset M$, $K = F(\gamma_1,\ldots,\gamma_n)$ and $K \subset M_n = M(\gamma_1,\ldots,\gamma_n)$, then $M_n = MK$, so
\begin{center}
$M \subset MK$ is a real radical extension.
\end{center}
\end{proof}

Exercise 8.6.2

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Exercise 8.6.2

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