Show that $G: Z[t] → \mathbb{C}$ defined by $G(f)=f(\sqrt{-1})$ is a ring homomorphism with kernel $(t^2 + 1)$ and image the Gaussian integers.

Question

I'm clear about the ring homomorphism and image part. But I'm not sure how to formulate my language precisely about the statement of the kernel.

Answer 1

First show that $\langle t^2+1 \rangle\subseteq \ker(G)$. This is easy as $G(t^2+1)=0$. So $t^2+1 \in \ker(G)$, hence the ideal generated by this is a subset of $\ker(G)$.

Now you need to show that $\ker(G) \subseteq \langle t^2+1 \rangle $. Let $f \in \ker(G)$, then $f(i)=0$. Since $f(t) \in \mathbb{Z}[t]$, thus $f(-i)=0$. This means $(x-i)(x+i)=x^2+1$ is a factor of $f$. Thus $f \in \langle t^2+1 \rangle$.

Answer 2

For the kernel statement I suggest

With the kernel being the principal ideal generated by $t^2 + 1$