# The number of ring homomorphism from $\mathbb Z[x,y]\to \mathbb{F}_2[x]/(1+x+x^2+x^3)$

Find the number of ring homomorphism from $$\mathbb Z[x,y]\to \mathbb{F}_2[x]/(1+x+x^2+x^3)$$.

My attempt: the ring $$Z[x,y]$$ has three generators $$1,x \ and\ y$$ we want $$1$$ to map to $$1.$$ Since the ring $$\mathbb{F}_2[x]/(1+x+x^2+x^3)$$ has 8 elements we have total of $$8\times 8$$ ring homomorphisms.

Am I right ?

It's fine, except that $$1$$ is not a generator of $$\Bbb Z[x,y]$$, only $$x$$ and $$y$$ are.
It's true, however, that the images of $$x,y$$ under a homomorphism $$f:\Bbb Z[x,y] \to R$$ can be arbitrary elements of $$R$$ and they uniquely determine $$f$$.
A ring homomorphism $$\varphi\colon R[x,y]\to S$$ (where $$R$$ and $$S$$ are any commutative rings) is completely determined by assigning
1. a ring homomorphism $$\varphi_0\colon R\to S$$,
2. $$\varphi(x)$$,
3. $$\varphi(y)$$.
In your case, $$R=\mathbb{Z}$$ and there exists a unique ring homomorphism $$\mathbb{Z}\to S$$, so you just need to assign $$\varphi(x)$$ and $$\varphi(y)$$, which makes for $$|\mathbb{F}_2/(1+x+x^2+x^3)|$$.