91Ó°ÊÓ

Find the fundamental unit for \(\mathbb{Q}(\sqrt{5}), Q(\sqrt{15}), \mathbb{Q}(\sqrt{2}), Q(\sqrt{3}), Q(\sqrt{624})\).

For which \(d\) does \(Q(\sqrt{d})\) have an integral basis of the form \(\alpha, \alpha\) ' where \(\alpha^{\prime}\) is the conjugate of \(\alpha\) ?

Let \(K\) be a quadratic number field with discriminant \(d\), and let \(\chi_{d}\) be the Kronecker symbol. Show, for \(p\) any prime, (a) \(p\) splits in \(K\) iff \(\chi_{d}(p)=1\). (b) \(p\) is inertial iff \(\chi_{d}(p)=-1\). (c) \(p\) ramifies iff \(\chi_{d}(p)=0\).