Question

4. Let $\phi: \mathbb{Z}[x] \to \mathbb{Z}[i]$ be a map defined by evaluating polynomials at $i = \sqrt{-1}$. Specifically, for $f(x) \in \mathbb{Z}[x]$, define: $\phi(f(x)) = f(i)$, where $\mathbb{Z}[i] = \{a + bi | a, b \in \mathbb{Z}\}$. (a) Prove that $\phi$ is a ring homomorphism. (b) Show that $\phi$ is surjective. (c) Prove that the kernel of $\phi$ is equal to the ideal generated by $x^2 + 1$. (d) Is the kernel of $\phi$ a prime ideal of $\mathbb{Z}[x]$? Justify your answer. Hint for (c): You may need to use the division algorithm. However, $\mathbb{Z}[x]$ is not a Euclidean domain. So how to use the division algorithm?

          4. Let $\phi: \mathbb{Z}[x] \to \mathbb{Z}[i]$ be a map defined by evaluating polynomials at $i = \sqrt{-1}$. Specifically, for $f(x) \in \mathbb{Z}[x]$, define:
$\phi(f(x)) = f(i)$,
where $\mathbb{Z}[i] = \{a + bi | a, b \in \mathbb{Z}\}$.
(a) Prove that $\phi$ is a ring homomorphism.
(b) Show that $\phi$ is surjective.
(c) Prove that the kernel of $\phi$ is equal to the ideal generated by $x^2 + 1$.
(d) Is the kernel of $\phi$ a prime ideal of $\mathbb{Z}[x]$? Justify your answer.
Hint for (c): You may need to use the division algorithm. However, $\mathbb{Z}[x]$ is not a Euclidean domain. So how to use the division algorithm?

4 let phi mathbbzx to mathbbzi be a map defined by evaluating polynomials at i sqrt 1 specifically for fx in mathbbzx define phifx fi where mathbbzi a bi a b in mathbbz a prove that phi is a 10636

Added by Travis H.

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