A partial field is an algebraic object that allows one to simultaneously abstract several different representability properties of matroids. In this paper we study partial fields as algebraic objects in their own right. We characterize the weak and strong characteristic sets of partial fields and show that the class of partial fields is not well-quasi ordered. We provide a new proof that the lift operator of a partial field is idempotent. We also provide a relation between the fundamental elements of a partial field and its Dowling lift, and show that the Dowling lift operator is idempotent.