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Divide two elements of $\mathbb{Z}[\sqrt{2}]$
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

In the ring $\mathbb{Z}[\sqrt{2}]$ , find the remainder $r=r_1+r_2\sqrt{2}$ , where $a \gt 0,\;b \gt 0$ , on dividing $a+b\sqrt{2}$ by $c+d\sqrt{2}$ .
Divide two numbers in $\mathbb{Z}[\sqrt{-2}]$
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

In the ring $\mathbb{Z}[\sqrt{-2}]$ , find the remainder $r=r_1+r_2\sqrt{-2}$ , where $a \gt 0,\;b \gt 0$ , on dividing $a+b\sqrt{-2}$ by $c+d\sqrt{-2}$ .