1. The problem statement, all variables and given/known data
Let [itex]K[/itex] be an algebraic field extension of a field [itex]F[/itex], and let [itex]L[/itex] be a subfield of [itex]K[/itex] such that [itex] F \subseteq L[/itex] and [itex]L[/itex] is normal in F. Show that if [itex]\sigma[/itex] is an automorphism of [itex]K[/itex] over [itex]F[/itex], then [itex]\sigma(L)=L[/itex].
2. Relevant equations
3. The attempt at a solution
I've been thinking about this for a while, but I couldn't really prove anything. So I'm just looking for a hint that will help me get started...
Let [itex]K[/itex] be an algebraic field extension of a field [itex]F[/itex], and let [itex]L[/itex] be a subfield of [itex]K[/itex] such that [itex] F \subseteq L[/itex] and [itex]L[/itex] is normal in F. Show that if [itex]\sigma[/itex] is an automorphism of [itex]K[/itex] over [itex]F[/itex], then [itex]\sigma(L)=L[/itex].
2. Relevant equations
3. The attempt at a solution
I've been thinking about this for a while, but I couldn't really prove anything. So I'm just looking for a hint that will help me get started...
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