Why is $\mathbb F_2[X]/X^3$ not isomorphic to $F_2[X]/X^3+2X+1$ $F_2[X]/ \langle X^3+2X+1\rangle$ has elements $0,1,x,x^2,1+x,1+x^2,x+x^2,1+x+x^2$
So does $F_2[X]/ X^3$? Why are these rings not isomorphic to each other?

The first one has a unique maximal ideal $(x)$. (Do you need help showing this?)
In the second one, the polynomial factors into irreducibles like this: $(x-1)(x^2+x+1)$, so there are two maximal ideals $(x-1)$ and $(x^2+x+1)$ in the ring.

Using the Chinese Remainder Theorem, you could go on to show the second ring is isomorphic to $F_2\times F_4$, which has no nonzero nilpotent elements, whereas the first ring has a nonzero element cubing to $0$.
