Forcing in model theory is a recent development in the metamathematics of algebra. The context of this development has three principal features: the importance of algebraically closed fields in commutative algebra and the existence of analogues of algebraically closed fields for other algebraic systems, earlier work on model-completeness and model-completions by Abraham Robinson and others, and Paul Cohen's forcing techniques in set theory. Algebraically closed fields serve a useful function in commutative algebra, algebraic number theory, and algebraic geometry. Certain arithmetical questions can be settled conclusively in an algebraically closed field. Examples are well-known. For instance, a system of polynomials has a common zero in some extension of their coefficient field if and only if they have a common zero in the algebraic closure of their coefficient field. In algebraic number theory, the study of
the prolongations of a valuation from its base field to a finite dimensional extension field reduces to the consideration of the embeddings of the extension field into the algebraic closure of the completion of the base field. A third example is the use of universal domains in algebraic geometry as the proper setting for the study of algebraic varieties over fields. In these and other instances, the existence and use of algebraically closed fields simplify the treatment of many mathematical problems.

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