Suppose that $F$ is a finite field of odd characteristic.

Suppose that $q(X_1,X_2,...,X_n)$ is a quartic (homogeneous) form with coefficients in $F$ such that

• $q$ is irreducible over $F$
• $q$ does not have any non-trivial zeros in $F^n$ (hence $n \leq 4$)
• over $\overline{F}$, $q$ can be factored as the product of four linear forms

What can one say about $q$? Can we say that $q$ must factor over the degree 4 extension of $F$?