Does there exist a finite field and 3-variate quadratic such that all non zero values of input give a non zero value of output

Question

I would like either an example of the following or a proof that no such things exists:

I would like a finite field F and homogeneous quadratic function in three variables $g(x,y,z)$ such that $g(x,y,z)=0$ implies $(x,y,z) = (0,0,0)$.

Another formulation:
Does there exist a finite field F and $3×3$ matrix A st. $v^TAv=0$ implies $v=0$ $\forall v \in F$?

Answer 1

On a finite field, it is impossible for an homogenous polynomial of degre 2 on 3 variables to vanish on only one point. Indeed such a polynomial would satisfy the condition of the Chevalley-Warning theorem (more variables than degree) and this theorem asserts that the number of zeros of this polynomial is a multiple of $p$ (the caracteristic). And there is no field with one element ... yet ! :)