The prime subfield is the intersection of all subfields i.e. it is the smallest subfield. If is a subfield then it means but then because is closed under addition and it has and it has all additive inverses. Therefore must be contained in . However, is a field and so for all . Therefore, the set of fraction is contained in , but this set of fractions is itself a field and so is in fact the set of all these fractions that we can form.