Roots of polynomials over finite fields

I've been trying to find the decomposition of $x^2-2$ to irreducible polynomials over $\mathbf{F}_5$ and $\mathbf{F}_7$.
I know that for some $a$ in $\mathbf{F}_5$ (for example), $x-a$ divides $x^2-2$ iff $f(a) = 0$, i.e $a$ is a root of $x^2-2$.
Over the field $\mathbf{F}_7$, I've found (by trail and error) that one irreducible polynomial is $x-3$.
I've now got two questions -

1. How can I find the other irreducible polynomial?


2. Is there any more efficient method to find roots than trial and error?

Thanks in advance