First-order characterization of function field invariants over large fields

Summary

Introduction

Definition 1.1 A field k is large if every smooth curve with a k-point has infinitely many k-points [Pop96, p. 2].

This condition is equivalent to the condition that k be existentially closed in the Laurent series field k((t)) [Pop96, Proposition 1.1]. It is in some sense opposite to the “Mordellic” properties satisfied by number fields, over which curves of genus greater than 1 have finitely many rational points [Fal83].

If p is any prime number, then any p-field (field for which all finite extensions are of p-power degree) is large [CT00, p. 360]. In particular, separably closed fields and real closed fields are large. Other examples of large fields include henselian fields and PAC fields. (PAC stands for pseudo-algebraically closed: a PAC field is one over which every geometrically integral variety has a rational point. See [FJ05, Chapter 11] for further properties of these fields.) For further examples of large fields, see [Pop96]. An algebraic extension of an large field is large [Pop96, Proposition 1.2].

Definition 1.2 Let k be a field. A function field over k is a finitely generated extension K of k with trdeg(K|k) > 0.

Definition 1.3 The constant field of a field K finitely generated over k is the relative algebraic closure of k in K.

Theorem 1.4There exists a formula φ(t) that when interpreted in a field K finitely generated over an large field k defines the constant field.