Irreducibility of $X^n - 2$ over $\mathbf Z[i]$.

Solution 1:

Let $\alpha=\sqrt[n]{2}$ then $L=\mathbb Q(\alpha)$ is a purely real extension of $\mathbb Q$. So we have that $i \notin L$ in particular $x^2+1$ is irreducible over $L$. It follows that $[L(i):L]=2$ and so $[\mathbb Q(i,\alpha):\mathbb Q(i)]=n$. Thereby $\alpha$ has degree $n$ over $\mathbb Q(i)$ so its minimum polynomial has degree $n$ which must be $x^n-2$. Since $x^n-2$ is irreducible over $\mathbb Q(i)$ it must be irreducible over its ring of integers, i.e. the Gaussian integers.

You'll notice that nothing here directly relates to $2$. We only needed that the extension $\mathbb Q(\alpha)$ was real of degree $n$. In particular this method demonstrates that if $x^n-k$ is irreducible over $\mathbb Z$ then it's irreducible over $\mathbb Z[i]$.