Exercise 13.3.3

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}}

In the proof of proposition 13.3.2, we asserted that $$\varphi (\alpha_1,...,\alpha_n)=\varphi (\alpha_{	au(1)},...,\alpha_{	au(n)})$$ follows from $\beta=\varphi (\alpha_1,...,\alpha_n) \in F$ and $	au \in G_f$. Prove this.

\begin{proof}
Let $\sigma \in \Gal(L/F)$ corresponding to $	au \in G_f$, so that  $\sigma(\alpha_i) = \alpha_{	au(i)}, \ i= 1,\ldots,n$.

Since F is fixed for $\sigma$, and $\beta = \varphi (\alpha_1,...,\alpha_n)  \in F$, $\beta = \sigma(\beta)$, so that
$$\varphi (\alpha_1,...,\alpha_n) = \sigma(\varphi (\alpha_1,...,\alpha_n)).$$ 
Moreover
\begin{align*}
\sigma(\varphi (\alpha_1,...,\alpha_n))&= \varphi(\sigma(\alpha_1),\ldots,\sigma(\alpha_n)\
&=\varphi (\alpha_{	au(1)},...,\alpha_{	au(n)}). 
\end{align*}
Therefore, $\varphi (\alpha_1,...,\alpha_n)=\varphi (\alpha_{	au(1)},...,\alpha_{	au(n)})$ for $\varphi (\alpha_1,...,\alpha_n) \in F$ and $	au \in G_f$.
\end{proof}

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}}

Exercise 13.3.3

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Exercise 13.3.3

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