Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 20G / b

The element $\omega$ satisfiesModulo both $2$ and $3$, this polynomial factors as $X(X-1)$. Henceandwhere the respective ideal norms are $2,2,3,3$.

Now $N(\omega )=12$, and $\omega$ lies in ${\mathfrak{p}}_2$ and ${\mathfrak{p}}_3$ but not in their conjugate primes. Similarly $N(2+\omega )=18$, while $2+\omega$ lies in ${\mathfrak{p}}_2$ and ${\mathfrak{q}}_3$. Comparing norms gives