inverses of fundamental units of real quadratic fields

Question

Let $K$ be a real quadratic field and let $\nu$ be its fundamental unit (notice that $\nu$ is uniquely determined in this case). I want to know whether or not it is always the case that $$\nu^{-1}=\pm\overline\nu.$$ I notice that this checks out for the first several quadratic fields (i.e. low values for square free $d\in\mathbb{N}$ with $K=\mathbb{Q}(\sqrt d)$), but is it always the case?

Answer 1

Yes, always the (Galois) norm (from number field $k$ to $\mathbb Q$) of a unit is a unit in $\mathbb Z$, so is $\pm 1$. In the case of a quadratic field extension, the norm of $\alpha$ is $\alpha\cdot \alpha^\sigma$ where $\sigma$ is the non-trivial (Galois) automorphism.