Determination of irreducible polynomials over a given field

[catcherintherye]

I am required to find all irreducible polynomials of the form xsquared + ax + b over the field F3, I have the 9 cases infront of me, i can see when something is reducible say xsquared is p(x)q(x) where p=x, q=x, but i have particular difficulty seeing when something is irreducible, e.g i know that xsqd + x + 2 is but i don't know how to show it, just as i do not know how xsqd + x +1 is irreducible over F2, although i can see how xsqd + 1 is since xsqd +1 = xsqd + 2x + 1 =(x+1)sqd, how do i know that a similar trick could not have been employed to make xsqd + x + 1 factorise?

 

[matt grime]

It's just a quadratic. If it is reducible then the factors are linear. So it is reducible if and only if one of 0,1,2 is a solution.