I want to show that $X^3-8X+16$ is irreducible over $\mathbb{Q}(i)$. I'd like to know what the most efficient way to do this would be (i.e. shortest way to write something that works down in a way that is also clear).
Here's my method (with gory details removed): suppose $\alpha$ is a root. Then $|\alpha|^3 \leq 8|\alpha| + 16$. Suppose $|\alpha| \geq 1$, then $|\alpha|^3 \leq 24 | \alpha |$. So $|\alpha| < \sqrt{24} \approx 4.89$. Check all values in $\mathbb{Z}[i]$ with modulus less than $\sqrt{24}$ and confirm they are not roots. Thus the polynomial is irreducible over $\mathbb{Z}[i]$. Thus by Gauss' lemma, it is irreducible over $\mathbb{Q}(i)$.
Can someone do better? My proof takes a while since there's a lot of values in $\mathbb{Z}[i]$ with modulus lower than $\sqrt{24}$ to check.