Why is $X^2+1$ reducible in $F_2$? One of my modules contains parts of ring theory which I have not done in over a year. So I appreciate this question may seem very basic as I am giving myself a crash course in ring theory by studying examples. 
Why is $X^2+1$ reducible in $F_2$?
And why is $X^2-X+1$ irreducible in $F_2$?

It is reducible since it has a root in the field...Note : don't use the converse that if a polynomial doesn't have a root in the field  means it is irreducible.This is not the case always.But is true only if you have a polynomial of degree 2 or 3.
