Positivity preservers over finite fields II
We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified for every $n\geq 2$, with one remaining case: $n=2$, $q\equiv 1\pmod 4$, and $q$ not a square. We settle this case by proving that every positivity preserver on $M_2(\mathbb{F}_q)$ is injective on the set $\mathbb{F}_q^+$ of nonzero squares whenever $q\equiv 1\pmod 4$. The proof combines an idempotent reduction of positivity preservers with a well-known property of quadratic characters. This yields the complete classification of entrywise positivity preservers over every finite field and in every fixed dimension.