Celebratio Mathematica — van den Dries

Complete Bibliography

Works connected to Gregory L. Cherlin

Filter the Bibliography List

year	title	people

			clear

G. Cherlin, L. van den Dries, and A. Macintyre: “Decidability and undecidability theorems for PAC-fields,” Bull. Am. Math. Soc. (N.S.) 4 : 1 (1981), pp. 101–104. MR 590820 Zbl 0466.12017 article

A pseudo-algebraically closed field (PAC-field for short) is a field \( K \) such that every absolutely irreducible variety defined over \( K \) has \( K \)-rational points. In [1968] Ax gave a decision method for the (elementary) theory of finite fields, the basis of which was his characterization of the infinite models of that theory as the perfect PAC-fields \( K \) with \[ G(K)\cong \hat{\mathbf{Z}} .\] (Here and in the following we use the notations \( G(K) \) for the absolute Galois group \[ \mathrm{Gal}(\tilde{K}\vert K) \] of \( K \), \( \tilde{K} = \) algebraic closure of \( K \), \( G = \) profinite completion of the discrete group \( G \).) In [1969] M. Jarden gave another natural source of PAC-fields: let \( e\in\mathbf{N} \); then for almost all \( e \)-tuples \[ (\sigma_1,\dots,\sigma_e)\in (\mathbf{Q})^e \] — in the sense of the Haar measure on \( G(\mathbf{Q}) \) — the fixed field \[ \mathrm{Fix}(\sigma_1,\dots,\sigma_e)\subset \tilde{\mathbf{Q}} \] is PAC, and \( e \)-free (where we call a field \( K \) \( e \)-free if \[ G(K)\cong \hat{F}_e ,\] \( F_e = \) free group on \( e \) generators). In 1975 Jarden and Kiehne classified \( e \)-free perfect PAC-fields up to elementary equivalence and derived the decidability of the theory of \( e \)-free perfect PAC-fields, cf. [1975]. The next step was an extension of these results to a certain class of PAC-fields with infinitely generated absolute Galois group, cf. [Jarden 1976]. We announce here solutions to the main questions provoked by this development.

Gregory L. Cherlin	Related
Angus Macintyre	Related