1. Objectives and summary. Much of elementary differential algebra can be regarded as a generalization of the algebraic geometry of polynomial rings over a field to an analogous theory for rings of differential polynomials (d.p.) over a differential field('). To date, however, considerable parts of basic algebraic geometry have yet to be "lifted" into differential algebra. The purpose of the present paper is to fill one such conspicuous gap by developing fundamental parts of a theory of specializations and dimensions over differential fields. Chapter I is devoted to certain necessary preliminaries. Among the concepts introduced is a useful weakening of the notion of reducedness of d.p. In terms of this, a type of set of d.p., called a coherent autoreduced set, is defined, for which a certain close relationship holds between the ideal and the differential ideal (d.i.) generated by the set. Coherent autoreduced sets of d.p. figure centrally in the proofs of the main theorems, since it turns out that their use enables one to reduce these theorems to analogous theorems for suitable polynomial rings. Chapter II contains the proofs of two basic theorems on extensions of specializations over differential fields. Roughly stated, these are:
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