Are $F_2[x]/\langle x^2\rangle$ and $F_2[x]/\langle x^2+x\rangle$ isomorphic?

Question

Are $F_2[x]/\langle x^2\rangle$ and $F_2[x]/\langle x^2+x\rangle$ isomorphic? I know they both have $4$ elements: $\{0,1,x,x+1\}$, but how do I define an isomorphism between them?

Answer 1

Hint:

$\mathbb F_2[x]/\langle x^2\rangle$ has nonzero nilpotent elements.

$\mathbb F_2[x]/\langle x^2+x\rangle \cong \mathbb F_2[x]/\langle x\rangle \times \mathbb F_2[x]/\langle x+1\rangle \cong \mathbb F_2 \times \mathbb F_2$ and so has no nonzero nilpotent elements.

It is instructive to see how their multiplication tables differ.