A fact about algebraic integers in a quadratic field

Let D be a squarefree integer. Prove that \{ a+b\sqrt{D} \ |\ a,b \in \mathbb{Z}\} \cup \{ (m + \frac{1}{2}) + (n + \frac{1}{2})\sqrt{D} \ |\ m,n \in \mathbb{Z} \} = \{ c + d(1 + \sqrt{D})/2 \ |\ c,d \in \mathbb{Z}\}.


Let \omega = (1+\sqrt{D})/2.

(\subseteq) Evidently, a+b\sqrt{D} = a-b + b\omega, where a,b \in \mathbb{Z}, and (m + \frac{1}{2}) + (n+\frac{1}{2})\sqrt{D} = m-n + (n+1)\omega.

(\supseteq) If d = 2k, then evidently c+d\omega = c+k + k\sqrt{D}. If d = 2k+1, then evidently c+d\omega = c+k+\frac{1}{2} + (k + \frac{1}{2})\sqrt{D}.

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