Given a minimum polynomial for $\alpha$, find the minimum polynomial for $\alpha/3$

Question

Let $f(x)=x^3+2x^2+x+27$. Suppose $\alpha\in\mathbb{C}$ satisfies $f(\alpha)=0$, and let $K=\mathbb{Q}(\alpha)$.

1. Is $f(x)$ irreducible?
2. What is $[K:\mathbb{Q}]$?
3. What is $N_K(\alpha)$?
4. Is $\frac{\alpha}{3}\in\mathcal{O}_K$?
5. What is the minimum polynomial of $\frac{\alpha}{3}$?

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I've answered all of these except for #5. $f(x)$ is irreducible by taking $f$ mod $2$ and observing there are no linear factors (which suffices as $f$ is cubic). Since $f$ is irreducible, then $[K:\mathbb{Q}]=3$. By definition of the field polynomial $\prod_{i=1}^n(t-\sigma_i(\alpha))$ , $N_K(\alpha)=-27$. Since norms split over products, then $N_K(\frac{\alpha}{3})=\frac{1}{27}N_K(\alpha)=-1$, so $\frac{\alpha}{3}\in\mathcal{O}_K$.

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What I thought about doing for #5 is to scale the original minimum polynomial appropriately using $\frac{\alpha}{3}$, i.e. $g(x)=27x^3+18x^2+3x+27$, but this cannot be correct as minimum polynomials are monic. I'm stuck here.

Answer 1

Take your $g(x)$ and divide it by $27$, thereby getting $x^3+\frac23x^2+\frac19x+1$, and you're done!