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Let $K \leq E \leq F$ be fields such that $[E:K]=n$. Let $Aut_K F$ act on the set $S$ of intermediate fields where $\sigma(I)$ gives the action of $\sigma \in Aut_K F$ on an intermediate field $I$. Show that the orbit of $E$ has at most $n$ elements.

Is it really obvious that $Aut_K F$ is a subset of the stabilizer of $E$? My intuition fails me here.

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    There seems to be a contradiction here. Since the stabilizer is a subgroup of $Aut_K F$, if the inclusion you have to show holds, the two sets are equal. Which in turn means that every $K$-automorphism of $F$ fixes $E$, which is not true in general.2012-08-11

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As shown in the comments, there is a problem with the question you are asking. However, about the question in the shaded area, I suggest you have a look at Theorem 3.8 in this note by Keith Conrad.