On Some Elementary Invariants of Fields

In this chapter we wish to review a number of classical properties and invariants of fields, and to discuss their elementary (or non-elementary) nature. Actually, we wish to distinguish between two notions of elementary properties of fields. The first, and weaker notion, is that of a property P of fields (in the usual, wide sense of non-formalized mathematics) such that whenever a field F has the property P , so does any other field F ′ which is elementarily equivalent to F . The second, and stronger notion, is that of a proprerty P of fields which can be given by a sentence φ in the language of rings. We call the first property simply elementary, or an elementary invariant, and the second property finitely axiomatizable. By way of explanation, observe that if P is an elementary property, then it can be axiomatized – it is equivalent to a possibly infinite union of first-order sentences φ – indeed the definition of an elementary property ensures that the collection of sentences true in every field having property P is well-defined, and this is the axiomatization. If the axioms are themselves equivalent to a finite list, they are equivalent to a single sentence φ – this justifies our terminology.