When are kernels of homomorphisms on $\mathbb{Z}[x]$ principal ideals, and what does this have to do with Gauss's lemma? Two questions listed in Artin under a section called "Gauss's lemma":


*

*Are the kernels of $f(x) \mapsto f(1 + \sqrt{2})$ and $f(x) \mapsto f(\frac12 + \sqrt2)$ (for $f \in \mathbb{Z}[x]$) principal ideals?

*For $\alpha \in \mathbb{C}$, prove the kernel of $f(x) \mapsto f(\alpha)$ is principal.
What's the general approach for solving such problems? And what (if anything) does it have to do with Gauss's lemma (products of primitive polynomials are primitive)?
For 1, clearly $(x-1)^2 - 2 = x^2 - 2x - 1$ is in the kernel of the first map, and this polynomial is primitive, but I'm unsure how this helps in figuring out if the kernel is principal?
 A: 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.
A: I have no idea for the second problem, but for the first one I think that you can use Euclidean division algorithm to show that the kernel of that map is a principal ideal. It's true that you don't have Euclidean division algorithm for polynomial division over $\mathbb{Z}[x]$ in general, but $x^2-2x-1$ is a monic polynomial. So, you can still use Euclidean algorithm.
OK. I think Gauss' lemma helps us here: If a primitive polynomial can be factored over $\mathbb{Q}[x]$ then it should also be factored over $\mathbb{Z}[x]$.
