(Solved): 3.5. We have already seen the ring of Gaussian integers \( \mathbb{Z}[i] \). More generally, for a ...

3.5. We have already seen the ring of Gaussian integers \( \mathbb{Z}[i] \). More generally, for any integer \( D \) that is not the square of an integer, \( { }^{13} \) we can form a ring \[ \mathbb{Z}[\sqrt{D}]=\{a+b \sqrt{D}: a, b \in \mathbb{Z}\} . \] If \( D>0 \), then \( \mathbb{Z}[\sqrt{D}] \) is a subring of \( \mathbb{R} \), while if \( D<0 \), then in any case it is a subring of \( \mathbb{C} \). (a) Let \( \alpha=2+3 \sqrt{5} \) and \( \beta=1-2 \sqrt{5} \) be elements of \( \mathbb{Z}[\sqrt{5}] \). Compute the quantities \[ \alpha+\beta, \quad \alpha \cdot \beta, \quad \alpha^{2} . \] (b) Prove that the map \[ \phi: \mathbb{Z}[\sqrt{D}] \longrightarrow \mathbb{Z}[\sqrt{D}], \quad \phi(a+b \sqrt{D})=a-b \sqrt{D}, \] is a ring homomorphism. (For notational convenience, if \( \alpha=a+b \sqrt{D} \in \mathbb{Z}[\sqrt{D}] \), then people often write \( \bar{\alpha}=a-b \sqrt{D} \), similar to the notation for complex conjugation.) (c) With notation as in (b), prove that \( \alpha \cdot \bar{\alpha} \in \mathbb{Z} \quad \) for every \( \alpha \in \mathbb{Z}[\sqrt{D}] \)