Equation with norms of cyclic extensions of coprime degrees

Question

Let $\mathbb{K}$ be a quadratic extension of $\mathbb{Q}$ and $\mathbb{L}$ be a cyclic extension of $\mathbb{Q}$ of odd degree. Given a rational $r\neq 0$, does there always exist $k\in \mathbb{K}^*$ and $l \in \mathbb{L}^*$ such that $$N_{\mathbb{K}}(k)= rN_{\mathbb{L}}(l) \,\,?$$

If not, would this kind of equation verify a Hasse principle ?

Answer 1

Yes, this follows from the fact that the degrees of $\mathbb{K}$ and $\mathbb{L}$ are coprime.

The result follows from a simple application of Bezout's identity and the fact that, for an extension $k$ of $\mathbb{Q}$ of degree $n$ and $x \in \mathbb{Q}$, we have $N_k(x) = x^n$.