We show thatderandomizing polynomial identity testing over an arbitrary finitefield implies that NEXP does not have polynomial size booleancircuits. In other words, for any finite field F(q) of size q,PITqNSUBEXPNEXPPpoly , wherePITq is the polynomial identity testing problem over F(q), andNSUBEXP is the nondeterministic subexpoential time class oflanguages. Our result is in contract to Kabanets and Impagliazzo'sexisting theorem that derandomizing the polynomial identity testingin the integer ring Z implies that NEXP does have polynomialsize boolean circuits or permanent over Z does not have polynomialsize arithmetic circuits.