New characterization of the norm of the fundamental unit of $\mathbb{Q}(\sqrt{M})$

We give an elementary criterion for the norm of the fundamental unit ${\epsilon }_K$ of $K=\mathbb{Q}(\sqrt{M})$, $M$ square-free. More precisely, if ${\epsilon }_K=a+b\sqrt{M}$, $a,b\in \mathbb{Z}$ or $\frac{1}{2}\mathbb{Z}$, its norm ${\mathbf{S}}_K$ only depends on $m:=\mathbf{g}\mathbf{c}\mathbf{d}(\frac{a+1}{\mathbf{g}\mathbf{c}\mathbf{d}(a+1,b)},M)$ and $m^{\mathrm{\prime }}:=\mathbf{g}\mathbf{c}\mathbf{d}(\frac{a-1}{\mathbf{g}\mathbf{c}\mathbf{d}(a-1,b)},M)$ as follows when $-1$ is a global norm: ${\mathbf{S}}_K=-1$ if and only if $m=m^{\mathrm{\prime }}=1$ (resp. $m=m^{\mathrm{\prime }}=2$) for $M$ odd (resp. even) (Theorems 1.1 or 2.4).