Perfect PAC Fields of Bounded Corank

Abstract

This Chapter treats the theory of perfect PAC fields of corank bounded by e that contain a fixed countable Hilbertian field K. Although the models of this theory of the form $$K\left( {{\sigma _1}...,{\sigma _e}} \right)$$ which are not e-free are indexed by a set σ ∈ G (K) e of measure zero, they nevertheless determine the theory (Theorem 21.10). In analogy to the theory of perfect e-free PAC fields containing K (Section 18.5), each sentence is equivalent modulo the theory of perfect PAC fields of corank ≤e to a basic sentence (as in Proposition 18.21). If K has elimination theory, this leads to a recursive decision procedure for the theory.