Factorization in Z[\sqrt{\alpha}]

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Factorization in $\mathbb{Z}\left[\sqrt{\alpha}\right]$

This page plots the numbers $\le n$ and their different decompositions into irreducible elements in $\mathbb{Z}\left[\sqrt{\alpha}\right]$ (where $\alpha$ is an integer).
Can you identify for which $\alpha$ the factorization is unique?

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Factorization in Z[√α]\mathbb{Z}\left[\sqrt{\alpha}\right]

Factorization in $\mathbb{Z}\left[\sqrt{\alpha}\right]$