Exercise 7.2.9

Question

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Let $F \subset L$ be Galois, and suppose that $F \subset K \subset L$ is an intermediate field. The goal of this exercise is to show that the number of conjugates of $K$ in $L$ is
$$[\Gal(L/F):N] = \frac{|\Gal(L/F)|}{|N|},$$
where $N$ is the normalizer of $\Gal(L/K)$ in $\Gal(L/F)$. More precisely, suppose that the distinct conjugates of $K$ are
$$K = \sigma_1 K,\sigma_2 K,\ldots,\sigma_r K,$$
where $\sigma_1 = e$. Then we need to show that $r = [\Gal(L/F):N]$.
\be
\item[(a)] Show that $\Gal(L/F)$ acts on the set of conjugates $\{\sigma_1 K,\sigma_2 K,\ldots,\sigma_r K\}$.
\item[(b)] Show that the isotropy subgroup of $K$ is the normalizer subgroup $N$.
\item[(c)] Explain how $r = [\Gal(L/F):N]$ follows from the Fundamental Theorem of Group Actions (Theorem A.4.9 from Appendix A).
\ee

Richard Ganaye · Accepted Answer

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\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

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ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

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\begin{proof}
\begin{enumerate}
\item[(a)] 
Write $O = \{\sigma_1 K, \sigma_2 K,\cdots,\sigma_r K\}$ the set of conjugate fields of $K$ and  $r = \vert O \vert $.

If $\sigma \in \Gal(L/F)$, and $M = K_j = \sigma_j K \in O,1\leq j \leq r$, write $\sigma \cdot M = \sigma M = \sigma K_j$ : 
$$\sigma\cdot K_j = \sigma \cdot (\sigma_j K) = (\sigma \circ \sigma_j) K.$$
Therefore $\sigma \cdot M = \sigma\cdot K_j$ is a conjugate field of $K$, so 
$$M \in O \Rightarrow \sigma \cdot  M\in O.$$

Moreover, for all $M \in O , e \cdot M = e M = M$, and if $\sigma,	au \in \Gal(L/F), \sigma \cdot (	au \cdot M) = \sigma(	au M) = (\sigma \circ 	au)M = (\sigma\circ 	au)\cdot M$.

So $G = \Gal(L/F)$ acts on the set $O =  \{\sigma_1 K, \sigma_2 K,\cdots,\sigma_r K\}$ of the conjugate fields of $K$, the action being defined by $\sigma \cdot M= \sigma M \ (\sigma \in \Gal(L/F), M \in O)$.

\item[(b)] Let $G_K$ the stabilizer of $K$ for this action : $G_K = \{\sigma \in G \ \vert \ \sigma K =K\}$.

By Exercise 7, for all $\sigma \in G = \Gal(L/F)$, 
$$\sigma K = K \iff \Gal(L/K) = \sigma \Gal(L/K) \sigma^{-1} \iff \sigma \in N.$$

Thus $G_K = N$.

\item[(c)]
The orbit ${\cal O}_K$ of $K$ for the action of $G = \Gal(L/F)$ on $O$ is  the whole $O$, since $O$ is by definition the set of conjugate fields of $K$: ${\cal O}_K = O$. the Fundamental Theorem of Group Actions gives then the equality
$$r = \vert {\cal O}_K \vert = [G : G_K] = [\Gal(L/F) : N].$$
The number of distinct conjugate fields of $K$ is so the index $[G : N_G(H)]$ of the normalizer of $H = \Gal(L/K)$ in $G= \Gal(L/F)$.

\end{enumerate}
\end{proof}

Exercise 7.2.9

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Exercise 7.2.9

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