This chapter deals with differential polynomials. It first presents some basic facts about differential fields that are of characteristic zero with one distinguished derivation, along with their extensions. It then considers various decompositions of differential polynomials in their natural setting, along with valued differential fields and the property of continuity of the derivation with respect to the valuation topology. It also discusses the gaussian extension of the valuation to the ring of differential polynomials and concludes with some basic results on simple differential rings and differentially closed fields. In contrast to the corresponding notions for fields, the chapter shows that differential fields always have proper d-algebraic extensions, and the differential closure of a differential field K is not always minimal over K.
Keywords: differential polynomial, differential field, decomposition, valued differential field, continuity, derivation, valuation topology, gaussian extension, simple differential ring, differentially closed field
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