Specifically, a field is a commutative ring in which every nonzero element is assumed to have a multiplicative inverse. Some themes of field theory are then immediately apparent. First, the study of fields is the appropriate venue for the consideration of some topics in number theory. Likewise one might include in Field Theory topics in local fields or finite fields, but the current Mathematics Subject Classification places these with Number Theory. The field of meromorphic functions on a given complex domain comes equipped with an additional operator, differentiation.
Differential algebra is the study of such fields, suitably axiomatized.
In addition, it is possible to consider differential equations over other fields than the real and complex numbers, leading to p -adic differential equations. A different additional structure which may be placed on a field is a topology, and in particular a metric, a valuation, or an ordering. For Galois theory applied specifically to algebraic number fields, see especially sections 11R32 and 11S20. Algebraic Geometry , although these are really just discussions of fields of finite transcendence degree over the ground field. Euclidean constructions .