of characteristic zero the rationals are always a subfield of the field of constants of . Any ring is a differential ring with respect to the trivial derivation which maps any ring element to zero. The field has a unique structure as a differential field, determined by setting the field axioms along with the axioms for derivations ensure that the derivation is differentiation with respect to For example, by commutativity of multiplication and the Leibniz law one has that The differential field fails to have a solution to the differential equation but expands to a larger differential field including the function which does have a solution to this equation. A differential field with solutions to all systems of differential equations is called a differentially closed field. Such fields exist, although they do not appear as natural algebraic or geometric objects. All differential fields (of bounded cardinality) embed into a large differentially closed field. Differential fields are the objects of study in differential Galois theory. Naturally occurring examples of derivations are partial derivatives, Lie derivatives, the Pincherle derivative, and the commutator with respect to an element of an algebra. Weyl Algebra Every differential ring has a naturally associated Weyl algebra , which is a noncommutative ring where and satisfy the relation . Such modules are called D-modules. In particular itself is a -module. All -ideals in are -submodule. For a differential rings there is an embedding of the Weyl algebra in the ring of pseudodifferential operators as the finite tails of these infinite series. Ring of pseudo-differential operators In this ring we work with which is a stand-in for the integral operator. Differential rings and differential algebras are often studied by means of the ring of pseudo-differential operators on them. This is the set of formal infinite sums where means that the sum runs on all integers that are not greater than a fixed (finite) value. This set is made a ring with the multiplication defined by linearly extending the following formula for "monomials": where is the binomial coefficient. (If the sum is finite, as the terms with are all equal to zero.) In particular, one has for and and using the identity See also − a differential algebra with an additional grading. − an algebraic structure with several differential operators acting on it. References As PDF External links David Marker's home page has several online surveys discussing differential fields.