From Tarski’s 1929 proof of the decidability of the first-order theory of the field of real numbers and Robinson’s 1949 proof of the complementary theorem that theory of the rational numbers is undecidable, the boundary between theories of fields with decidable or undecidable theories has remained unclear. Many more examples are known on both sides of the divide. Generally, the proofs mix ideas from logic, arithmetic, and geometry. The notable test case of theory of the field of rational functions in one variable over the complex numbers was explicitly raised in print in 1963 but remains open to this day.

I will discuss the history of this problem, will describe some recent work, and will explain why its resolution has been difficult.