Exercise 9.7

Proof. :

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$(1-2\omega )\omega =2+3\omega \equiv 2(\mathit{mod}3)$, so $2+3\omega$ is primary, and associate to $1-2\omega$. $N(2+3\omega )=7$ and 7 is a rational prime, thus $2+3\omega$ is a primary prime.

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$-7-3\omega \equiv 2(\mathit{mod}3)$.

$N(-7-3\omega )=37$ and 37 is a rational prime, thus $-7-3\omega$ is a primary prime.

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$(3-\omega ){\omega }^2=-4-3\omega \equiv 2(\mathit{mod}3)$, so $-4-3\omega$ is primary, and associate to $3-\omega$.

$N(-4-3\omega )=13$ and 13 is a rational prime, thus $-4-3\omega$ is a primary prime.