I have a polynomial $p(u,v) \in F_q[u,v]$ which is symmetric. In case $q \equiv −1\pmod{10}$, by experiment, I understood that $p(u,v)$ is irreducible. But for every $u \in F_q$, $p_u(v)$ is product of a linear and two quadratic polynomials.Now I want to proof this. Can you introduce references that help me?
$p(u,v)=(u^3 + 1/2(-5t + 11)u^2)v^5 + (1/2(t + 1)u^4 + 1/2(3t + 29)u^3 + 1/2(-89t + 197)u^2 + (4t -9)u)v^4 + (u^5 + 1/2(3t + 29)u^4 + 1/2(-t + 151)u^3 + (-186t + 409)u^2 + 1/2(89t - 197)u + 1/2(-5t + 11))v^3 + (1/2(-5t + 11)u^5 + 1/2(-89t + 197)u^4 + (-186t + 409)u^3 + 1/2(t - 151)u^2 + 1/2(3t + 29)u - 1)v^2 + ((4t - 9)u^4 + 1/2(89t - 197)u^3 + 1/2(3t + 29)u^2 + 1/2(-t - 1)u)v + 1/2(-5t + 11)u^3 - u^2,$
where $t$ is $\sqrt 5 $ in $F_q$.