Assume that we have a field F, an extension field E of F, and both of them are contained in the algebraic clousure $\overline{F}$.
Let E have the property that every automorphism of $\overline{F}$ that leaves every element of F fixed, maps E onto itself.
Does then E have to be an algebraic extension of F?
(I am wondering about this because I think it can be show that a splitting field is an algebraic extension?, and I am unsure about a theorem in my book, if he sometimes just forgets to say that E is an algebraic extension, if it is not, or if it follows from something.)