Are the rings $\mathbb{Z}_2[X] /(X^2 + 1)$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ isomorphic?

Question

I'm struggling with the question. I know how to demonstrate that $\mathbb{R}[X]/(X^2 +1) \cong \mathbb{C}$ and similarly $\mathbb{R}[X]/(X^2 +2) \cong \mathbb{C}$. However I cannot come up with an understanding of the ring, or any candidate homomorphisms between $\mathbb{Z}_2[X]/(X^2 +1)$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$

I cannot figure out how to map the products.

What functions should I try testing?

Answer 1

The elements of $K= \mathbb Z_2[x]/(x^2+1)$ are $0,1,x,1+x$. As $(x+1)^2= x^2+1$ in $\mathbb Z_2[x]$, we get that $y^2=0$ in $K$ for $y=x+1$.

While no non zero elements of $\mathbb Z_2 \times \mathbb Z_2$ have a square equal to zero.

So the rings can’t be isomorphic.

Answer 2

Hint:

Does a product of fields have nonzero nilpotent elements?