How to use finite fields for problems concerning infinite fields

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The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.
12 pages; to appear in Proc.Conference Marseille-Luminy 2007, Contemp. Math. series, AMS, 2009

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