Field Arithmetic - Pseudo Algebraically Closed Fields

Pseudo Algebraically Closed Fields

A pseudo algebraically closed field (in short PAC) K is a field satisfying the following geometric property. Each absolutely irreducible algebraic variety V defined over K has a K-rational point.

Over PAC fields there is a firm link between arithmetic properties of the field and group theoretic properties of its absolute Galois group. A nice theorem in this spirit connects Hilbertian fields with ω-free fields (K is ω-free if any embedding problem for K is properly solvable).

Theorem. Let K be a PAC field. Then K is Hilbertian if and only if K is ω-free.

Peter Roquette proved the right-to-left direction of this theorem and conjectured the opposite direction. Michael Fried and Helmut Völklein applied algebraic topology and complex analysis to establish Roquette's conjecture in characteristic zero. Later Pop proved the Theorem for arbitrary characteristic by developing "rigid patching".