Some basic theorems in differential algebra (characteristic 𝑝, arbitrary)
1952; American Mathematical Society; Volume: 73; Issue: 1 Linguagem: Inglês
10.1090/s0002-9947-1952-0049174-x
ISSN1088-6850
Autores Tópico(s)Numerical methods for differential equations
Resumocannot be taken over verbatim. This usually requires that the two cases be discussed separately, and this has been done below. The subject is treated ab initio, and one may consider that the proofs in the case of characteristic 0 are being offered for their simplicity. In the field theory, questions of separability are also considered, and the theorem of S. MacLane on separating transcendency bases [4] is established in the differential situation. 1. Definitions. By a differentiation over a ring R is meant a mapping u-*u' from R into itself satisfying the rules (uv)'=uv'+u'v and (u+v)' =u'+v'. A differential ring is the composite notion of a ring R and a differentiation over R: if the ring R becomes converted into a differential ring by means of a differentiation D, the differential ring will also be designated simply by R, since it will always be clear which differentiation is intended. If R is a differential ring and R is an integral domain or field, we speak of a differential integral domain or differential field respectively. An ideal A in a differential ring R is called a differential ideal if uGA implies u'GA. The ring { u+A } of residue classes of the differential ring R mod a differential ideal A is also a differential ring under the differentiation (u+A)'=u'+A. If F is a differential field, then from (v.u/v)'=v(u/v)'+v'(u/v) we obtain (u/v)'=(u'v-uv')/v2. If R is a differential integral domain, then its quotient field F becomes a differential field on setting (u/v) ' = (u'v - uv') /V2: it is this differential field which is intended when we speak of the
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