First-order Characterization of Function Field Invariants over Large Fields

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 ppower 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].