Let $k$ be an arbitrary field, $d\in k$, and $d$ is not a square in $k$. Consider the binary quadratic form $f(x,y)=x^2-d y^2$ (it is the norm from $k(\sqrt{d})$ to $k$). I am looking for a reference to a proof of the following fact:

There exist a field extension $K/k$ and a non-zero element $a\in K$ such that $f$ does not represent $a$ over $K$ (that is, there is no $x,y\in K$ such that $a=x^2-d y^2$).

Edited question (which I really meant): Let $l/k$ be a separable quadratic field extension (of fields of any characteristic). Prove that there exists a field extension $K/k$ such that the norm map $N\colon (l\otimes_k K)^\times\to K^\times$ is not surjective.

Local Fieldsto the unramified quadratic extension $K(\sqrt{d})/K$. In particular, the element $t$ is not represented. – Pete L. Clark Jun 6 '10 at 14:07