Element $a$ generates a multiplicative group of a field F of $343$ elements. Is the polynomial $x^2+ax-a+2a^2$ irreducible in the polynomial ring $F[x]$?
As I understand, $F[x]$ is the ring of all polynomials, which can be generated by the coefficient from $F$. How then can I check a reducibility of a polynomial $p(x)$ in $F[x]$? Should I try to decompose it into the $p(x)=g(x)*h(x)$, where "=" means equality modulo 343?