Is there the standard name for a field extension like this?

Question

Let $E/F$ be a field extension such that $F$ is the fixed field of $Aut(E/F)$. Is there a standard name for this field extension? Hungerford calls this extension as "Galois extension" (which does not assume it to be algebraic) but it is more general than the Galois extension (normal separable) in standard sense. What would you call this extension? Does it sound okay to call "$E/F$ is a semi-Galois extension" or "$E/F$ is a pseudo-Galois extension" or something like that?

Answer 1

The terminology used by Hungerford seems reasonable, since if $ E/F $ is an algebraic extension that is Galois in this sense, then it is also Galois in the usual sense (normal and separable). Note that normality itself is not a notion that readily generalizes to transcendental extensions, since the usual definition of normality only makes sense for algebraic extensions (you can't embed transcendental extensions into an algebraic closure).