Let $d$ be a positive integer which is not a perfect square. We have the norm multiplicative group homomorphism, $N:{\mathbb Q}[\sqrt{d}] \to {\mathbb Q}$ defined by $N(x+y\sqrt{d})=x^2-dy^2$.
It is well-known that the restricted kernel $\lbrace z\in {\mathbb Z}[\sqrt{d}] \ \big| \ N(z)=1 \rbrace$ is an "infinite cyclic" group, i.e. it is generated by a single element.
What about the unrestricted kernel $\lbrace z\in {\mathbb Q}[\sqrt{d}] \ \big| \ N(z)=1 \rbrace$ ? Is it also finitely generated, and if so how many generators are needed to generate it ?