Differential Weil Descent and Differentially Large Fields

A differential version of the classical Weil descent is established in all characteristics. It yields a theory of differential restriction of scalars for differential varieties over finite differential field extensions. This theory is then used to prove that in characteristic 0, $\mathit{\text{differential}}\mathit{\text{largeness}}$ (a notion introduced here as an analogue to largeness of fields) is preserved under algebraic extensions. This provides many new differential fields with minimal differential closures. A further application is Kolchin-density of rational points in differential algebraic groups defined over differentially large fields.