Non-algebraic Galois Extension

Say $\displaystyle E/F$ is Galois*.

Must it be the case that $\displaystyle E/F$ is algebraic?

(This was not addressed in my book).

*)I just realized how bad my question is. A Galois extension $\displaystyle E/F$ is an algebraic extension such that $\displaystyle F = E^{\text{Gal}(E/F)}$. Note we use "algebraic" within the definition itself. My question would be more appropriately asked if $\displaystyle F = E^{\text{Gal}(E/F)}$ then must it follow that $\displaystyle E/F$ is algebraic?