yes! let be the minimal polynomial of i.e. is monic and it has the smallest degree among all with this property that then and, since

is surjective, we have finally, since is irreducible and is a PID, the ideal is maximal and thus is a field. as a result

the converse is also true: if is a field, then is algebraic over the reason is that if is transcendental over then and obviously is never a field.