## Prove that two given polynomials are irreducible over the Gaussian rationals

Show that $p(x) = x^3-2$ and $q(x) = x^3-3$ are irreducible over $F = \mathbb{Q}(i)$.

Recall that the Gaussian integers $\mathbb{Z}[i]$ are a Unique Factorization Domain, and that $\mathbb{Z}[i] \subseteq \mathbb{Q}(i)$. We claim that $\mathbb{Q}(i)$ is the field of fractions of $\mathbb{Z}[i]$. Indeed, if $F$ is a field containing $\mathbb{Z}[i]$, then $F$ also contains any Gaussian rational since we can write $\frac{a}{b} + \frac{c}{d}i = \frac{1}{bd}(ad + bci)$ with $a,b,c,d \in \mathbb{Z}$. By the universal property of fields of fractions, $\mathbb{Q}(i)$ is the field of fractions of $\mathbb{Z}[i]$.

Recall that $1+i$ and 3 are irreducible in $\mathbb{Z}[i]$. So Eisenstein’s criterion applies, showing that $p$ and $q$ are irreducible over $\mathbb{Q}(i)$.