Differential Modules over Differential Fields

Differential modules over differential fields for which the Picard–Vessiot differential Galois is available: characteristic zero differential fields F with an algebraically closed field of constants C. After reviewing the Galois theory, including the Picard–Vessiot (PV) closure and its Picard–Vessiot (PV) ring as well as the pro-algebraic differential Galois group G of PV closure over F. It is shown how the ring of the PV closure can be used to set up a duality between finite-dimensional differential F modules and finite dimensional (over C rational G modules. The chapter explains how trivializing the differential F modules (that is, solving the differential equations they represent) comes about.