(Solved): Please show all steps and explanation   2. Recall that the map \( \phi: x+y \sqrt{2} \mapst ...

Please show all steps and explanation

 

2. Recall that the map \( \phi: x+y \sqrt{2} \mapsto\left|x^{2}-2 y^{2}\right| \) is a Euclidean function on \( \mathbb{Z}[\sqrt{2}] \). (a) If \( \alpha=41-4 \sqrt{2} \) and \( \beta=7-3 \sqrt{2} \), use the algorithm given in class to find elements \( \gamma, \rho \in \mathbb{Z}[\sqrt{2}] \) such that \( \alpha=\gamma \beta+\rho \) and \( \phi(\rho)<\phi(\beta) \). (b) If \( \alpha=7+2 \sqrt{2} \) and \( \beta=7-2 \sqrt{2} \), find, in any way you wish, elements \( \gamma, \rho \in \mathbb{Z}[\sqrt{2}] \) such that \( \alpha=\gamma \beta+\rho \) and \( \phi(\rho)<\phi(\beta) \). You may use the algorithm given in class if you so choose. Note: The \( \beta \) here is different from the \( \beta \) in part (a) (and likewise for \( \alpha \), of course).