\((a+b\sqrt{2}) + (a'+b'\sqrt{2}) = (a+a') + (b+b')\sqrt{2} \in \left\{a+b\sqrt{2} : a, b\in \mathbb{Q} \right\}\)
\((a+b\sqrt{2}) \cdot (a'+b'\sqrt{2}) = (aa'+4bb') + (ab'+a'b)\sqrt{2} \in \left\{a+b\sqrt{2} : a, b\in \mathbb{Q} \right\}\)
Assoziativ-, Kommutativ und Distributivgesetz folgen unmittelbar aus den Verknüpfungen auf \(\mathbb{R}\)
\(0 = 0+0\sqrt{2}\in \left\{a+b\sqrt{2} : a, b\in \mathbb{Q} \right\}\)
\(1 = 1+0\sqrt{2}\in \left\{a+b\sqrt{2} : a, b\in \mathbb{Q} \right\}\)
\(-(a+b\sqrt{2}) = -a + (-b)\sqrt{2} \in \left\{a+b\sqrt{2} : a, b\in \mathbb{Q} \right\}\)
\(\frac{1}{a+b\sqrt{2}} = \frac{1}{a^2 - b^2}(a-b\sqrt{2}) \in \left\{a+b\sqrt{2} : a, b\in \mathbb{Q} \right\}\)