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 $F_2$?

$$\forall\;\text{ prime}\;\;p\;,\;\;(a+b)^p=a^p+b^p\pmod p\implies (x^2+1)=(x+1)^2\pmod2$$
As for $\;x^2-x+x^2+x+1\pmod 2\;$ : check this quadratic has not roots in $\;\Bbb F_2\;$ and is thus irreducible (because a polynomial of degree less than four is irredicible over a field iff it has no roots in that field)
