A sentence false in a field of characteristic $0$ but true in all fields of positive characteristic?

Solution 1:

The canonical solution is to use the fact that $x^2 + y^2 = -1$ has a solution in every finite field, which can be proved by a simple counting argument. That equation is solvable in any field of positive characteristic, because it is solvable in the prime subfield. But not in $\mathbb R$.

This is the same idea as the other answer, but the proof of solvability is easier than the 4-squares theorem.

For the general question of what can be done in first-order theory of rings to separate characteristic 0 and characteristic $p$, there are brilliant results on this (and several other matters) in Bjorn Poonen's paper at http://arxiv.org/abs/math/0507486 .

Solution 2:

The four square theorem yields the following theorem in positive characteristic:

$$\exists a,b,c,d : a^2 + b^2 + c^2 + d^2 + 1 = 0$$