Why does the definition for the fixed field of $H$ specify that $H$ must be a subgroup of the Galois group of the field extension? As far as I can see, if $H$ is simply a subset, the 'fixed field' of $H$ is still a field, as we have closure under all $4$ operations.
So, am I missing something, or is the requirement for $H$ to be a subgroup not necessary for the fixed field of $H$ to actually be a field?
If a field is fixed by a subset $S$ of $G$, it is also fixed by the subgroup generated by $S$, so there is no loss of generality.