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I'm having trouble with an algebraic number theory problem. Let $R = \mathbb{Z}[\sqrt{-3}]$. The problem is to show that $(2, 1 + \sqrt{-3})$ is the unique prime ideal containing the ideal $(2)$, and to conclude that $(2)$ can't be written as a product of prime ideals. Aside from actually solving the problem I don't even understand how proving this implies that $(2)$ can't be factored. Wouldn't we want to show that $(2)$ contains no prime ideals instead? Thanks.

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