Model Theory of Valued Fields

Abstract

This chapter is devoted to the model theory of valued fields, which is due to Ax and Kochen. We also present Ax–Kochen’s solution of Artin’s conjecture that for every prime p, the field of p-adic real numbers ${\mathbb Q}_p$ is a $C_2(d)$ field for every $d\ge 1$ (See [2–4]). This was probably the first occasion when model theoretic methods were used to solve an outstanding conjecture in mathematics. This chapter requires a good knowledge of valued fields. It is a specialised topic not commonly covered in graduate courses. In Appendix C, we have given a self-contained account of the theory of valued fields that we require. The reader not familiar with valued fields should go through Sect. C.1 before proceeding with this chapter.