Two questions listed in Artin under a section called "Gauss's lemma":
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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?
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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?
Best Answer
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.