On some elementary invariants of fields [expository notes]

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Field InvariantsOn some elementary invariants of fields. (pdf) (15 pages) These notes, mostly written after I attended the 2003 Arizona Winter School on model theory and arithmetic, give a sort of introduction to the model theory of fields (assuming, unfortunately, that you know some model theory and some arithmetic geometry and have somehow never managed to combine them!). The point of departure is the search for "invariants" of elementarily equivalent fields. Among other things, I point out that a standard conjecture relating period and index in the Brauer group would play the same role as the Milnor Conjecture did in defining the transcendence degree of an absolutely finitely generated field. But I picked a bad time to try an exposition on the model theory of fields: since this paper has been written, Scanlon has proven the equivalence of elementary equivalence and isomorphism for finitely generated fields of characteristic zero, and Poonen and Pop have presented much stronger definability results than the ones I discuss in Section 2. Perhaps someday I will update the exposition to include these exciting results.