Undecidability

In contrast to the theories considered so far (e.g. the theory of finite fields, the theory of almost all fields ℚ(σ₁,…,σ _e ) for fixed e, and the theory of perfect PAC fields of bounded corank), the theory of perfect PAC fields is undecidable. This is the main result of this chapter (Corollary 28.10.2). An application of Cantor’s diagonalization process to Turing machines shows that certain families of Turing machines are nonrecursive. An interpretation of these machines in the theory of graphs shows the latter theory to be undecidable. Finally, Frattini covers interpret the theory of graphs in the theory of fields. This applies to demonstrate the undecidability of the theory of perfect PAC fields.