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A fundamental discriminant is an integer that occurs as the field discriminant of a quadratic number field. For square-free $d$, it is $d$ when $d\equiv 1(\mathrm{mod}4)$ and $4d$ otherwise.

The ring $\mathbb{Z}[{\zeta }_3]$ is the full ring of integers of the quadratic number field $\mathbb{Q}(\sqrt{-3})$. Every ring of integers of a number field is a Dedekind domain, so this ring is Dedekind.