Celebratio Mathematica — van den Dries

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L. van den Dries and R. L. Smith: “Decidable regularly closed fields of algebraic numbers,” J. Symbolic Logic 50 : 2 (June 1985), pp. 468–475. MR 793127 Zbl 0574.12023 article

A field \( K \) is regularly closed if every absolutely irreducible affine variety defined over \( K \) has \( K \)-rational points. This notion was first isolated by Ax [1968] in his work on the elementary theory of finite fields. Later Jarden [1976] and Jarden and Kiehne [1975] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field \( K \) with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in \( K \). The method of proof employed in [Jarden 1972, 1976; Jarden and Kiehne 1975] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [1980] and to the forthcoming book by Fried and Jarden [1986] for a more thorough discussion of the latest results.