When are kernels of homomorphisms on $\mathbb{Z}[x]$ principal ideals, and what does this have to do with Gauss's lemma?

For the second case, consider for some $\alpha \in \mathbb{C}$, the evaluation map over the rationals, that is

$$ \operatorname{ev}_\alpha: \mathbb{Q}[x] \to \mathbb{C} \enspace \enspace \enspace f \mapsto f(\alpha). $$

Here, since we're working over a field, then the kernel is exactly the principal ideal $(m_\alpha(x))$ where $m_\alpha(x)$ is the minimal polynomial of $\alpha$. Note that this implies the map is injective if $\alpha$ is not in $\overline{\mathbb{Q}}$.

Now, $\mathbb{Z}[x] \subset \mathbb{Q}[x]$ so the kernel of the evaluation map restricted to $\mathbb{Z}[x]$ is just $(m_\alpha(x)) \cap \mathbb{Z}[x] = I$. We want to show this ideal is principal in $\mathbb{Z}[x]$. We can multiply through by the least common multiple of the denominators of the coefficients in $m_\alpha(x)$ to obtain a primitive polynomial $p_\alpha(x)$ with integer coefficients. I claim that $p_\alpha(x)$ generates $I$.

To see this, consider $f(x) \in I$. Now, $m_\alpha(x) \mid f(x)$ since $f(\alpha) = 0$. Thus, $\deg f \geq \deg m_\alpha$. But $\deg m_\alpha = \deg p_\alpha$. Now, using the Euclidian algorithm in $\mathbb{Q}[x]$, we can write $f(x) = p_\alpha(x) q(x) + r(x)$ where $q(x)$ and $r(x)$ have possibly rational coefficients and $\deg r < \deg p_\alpha$. However, $f$ and $p_\alpha$ vanish at $\alpha$ so $r(x)$ must also vanish at $\alpha$. This implies that either $m_\alpha(x) \mid r(x)$ or $r(x) = 0$. The former case contradicts $\deg r < \deg p_\alpha$, so $r = 0$.

Now, we know that $f(x) = p_\alpha(x) q(x)$ where $f$ and $p_\alpha$ have integer coefficients and $p_\alpha$ is primitive. From here, it is a routine application of Gauss' lemma to show that $q(x)$ has integer coefficients as well so that $f$ is in the principal ideal of $\mathbb{Z}[x]$ generated by $p_\alpha(x)$ and thus the kernel of the evaluation map is principal generated by $p_\alpha(x)$. I'll let you fill in this last step yourself since your question was how to reduce it to Gauss' lemma.