Let $E$ be a finite extension field of $F$. Let $A$ be a subset of $Gal(E/F)$. Let $\langle A\rangle$ be the subgroup generated by $A$. Is the fixed field of $A$ equal to the fixed field of $\langle A\rangle$ ?
I understand that the fixed field of $\langle A\rangle$ is a subfield of the fixed field of $A$ because $A$ is a subset of $\langle A\rangle$. Does the reverse inclusion hold?
Yes. If the fixed field of $A$ is $E/K/F = \bigcap_{\phi \in A} E^{\phi}$, then based on the fact that any element in $\langle A\rangle$ is some composition of elements fixing $K$ (that is, the composition of some elements of $A$), we know that any element of $\langle A\rangle$ will also fix $K$.