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The theory DF of differential fields. The signature is that of fields (0, 1, +, -, ×) together with a unary function ∂, the derivation. The axioms are those for fields together with \forall u\forall v\,\partial(uv) = u \,\partial v + v\, \partial u \forall u\forall v\,\partial (u + v) = \partial u + \partial v\ . For this theory one can add the condition that the characteristic is p, a prime or zero, to get the theory DFp of differential fields of characteristic p (and similarly with the other theories below). If K is a differential field then the field of constants k = \{u \in K : \partial(u) = 0\}. The theory of differentially perfect fields is the theory of differential fields together with the condition that the field of constants is perfect; in other words for each prime p it has the axiom: \forall u \,\partial(u)=0 \and p 1 = 0\rightarrow \exists v\, v^p=u (There is little point in demanding that the whole field should be perfect field, because in non-zero characteristic this implies the differential is 0.) For technical reasons to do with quantifier elimination it is sometimes more convenient to force the constant field to be perfect by adding a new symbol r to the signature with the axioms \forall u \,\partial(u)=0 \and p 1 = 0 \rightarrow r(u)^p=u \forall u \,\lnot \partial(u)=0\rightarrow r(u)=0. The theory of DCF differentially closed fields is the theory of differentially perfect fields with axioms saying that such that if f and g are differential polynomials and the separant of f is nonzero and g≠0 and f has order greater than that of g, then there is some x in the field with f(x)=0 and g(x)≠0. |
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