In an exercise of the book Algebraic Theory of Numbers by Samuel, one must show that--in the integer ring $\mathcal{O_k}$ of the number field extension $\mathbb{Q}(x)$ where $x^{3}=2$--the ideal $x\mathcal{O_k}$ is a prime ideal. As a hint the author says to consider what consider the decomposition of ideal $2\mathcal{O_k}$ is in $\mathcal{O_k}$.
But I do not know how to use the hint to show this assertion. Any help is appreciated. Thanks.
Since $x^3=2$, and $8=N(2)=N(x^3)=N(x)^3$, we have $N(x)=2$. But then $|\mathcal{O}_K/(x)|=2$, where $|\cdot |$ is cardinality as a set, hence the quotient is a ring of size $2$, and so must be a field. Since the quotient is an integral domain, the ideal generated by $x$ is prime.