I have a symmetric polynomial $p(x) = x^5+a_4(y)x^4 + a_3(y)x^3+a_2(y)x^2+a_1(y)x+a_0(y)$ over the ring $F_q[x,y]$ where $F_q$ is a finite field with $q$ elements.
With experiment, i know that in case $q=9 \pmod {10}$, for each $y \in F_q$, $p(x)$ has one root. Therefore for each $y \in F_q$, $p(x)$ is not irreducible in this case, I try to use the Galois's theory to proof that $p(x)$ has one root and this means that Galois group of $p(x)$ is $Id \times S_4$, Is this true?
My information on Galois's theory in the finite fields is limited, Please help me or introduce references that help me.
Thank you.