Counting mod n in pseudofinite fields

We show that in an ultraproduct of finite fields, the mod-n nonstandard size definable sets varies definably families. Moreover, if K is any pseudofinite field, then one can assign “nonstandard sizes mod n” to K. As n varies, these assemble into a strong Euler characteristic on K, taking values profinite completion \(\hat {\mathbb{Z}}\) integers. The not canonical, but depends choice Frobenius. When Abs(K) finite, has some funny properties for two choices Frobenius.Additionally, we theory fields remains decidable when first-order logic expanded with parity quantifiers. However, proof computational algebraic geometry statement whose deferred later paper.