Comparing two quadratic number fields Let $\;$$K = \Bbb Q$[$\sqrt{-5}$] $\;$and let$\;$ $L = \Bbb Q$[$\sqrt{-6}$]$\,$. While it is clear that, as fields, K and L are distinct, if each arithmetic operation is considered separately, they seem to share a number of common features.  For example, on the additive side, K and L are isomorphic since each is just a two dimensional vector space over the rationals.  On the multiplicative side, their respective rings of integers share the same trivial unit group (w = 2) and the same nontrivial class group (h = 2).  Can one say more?
Question:  Are the multiplicative groups of K and L isomorphic? 
Thanks.

No, the quadratic number fields are not isomorphic, see here for several proofs concerning possible isomorphism between $\mathbb{Q}(\sqrt{d})$ and $\mathbb{Q}(\sqrt{d'})$. Moreover their discriminant and the  Minkowski bound of their rings of integers is different. For $\mathcal{O}_K=\mathbb{Z}[\sqrt{-5}]$ it is $\frac{2}{\pi}\sqrt{20}$, and for $\mathcal{O}_K=\mathbb{Z}[\sqrt{-6}]$ it is $\frac{2}{\pi}\sqrt{24}$. Still, the class number is $2$ in both cases.
