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From Dummit & Foote pg. 229:

Let $D$ be a squarefree integer. It is immediate from the addition and multiplication that the subset $\Bbb{Z}[\sqrt{D}] = \{a + b \sqrt{D} | a,b, \in \Bbb{Z}\}$ forms a subring of the quadratic field $\Bbb{Q}(\sqrt{D})$ defined earlier. If $D = 1 \pmod 4$ then the slightly larger subset $\Bbb{Z}[\frac{1 + \sqrt{D}}{2}]$ is also a subring: closure under addition is immediate and [...] Define $\mathcal{O}_{\Bbb{Q}(\sqrt{D})} = \Bbb{Z}[\omega]$ where $\omega = \sqrt{D}, $ if $D = 2,3 \pmod 4$ and $\omega = \frac{1 + \sqrt{D}}{2}, $ if $D = 1 \pmod 4$.

Why did they split up the definition of $\omega$ like that into a conditional, why is it needed? Thanks.

Daniel Donnelly
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  • Because then this ring will always be integrally closed and thus a dedekind domain. This notion is very important in algebraic number theory, because it implies unique factorization of an ideal into prime ideals. See here:https://en.m.wikipedia.org/wiki/Dedekind_domain – Jef Jun 26 '15 at 12:31

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