Let $$ L=\mathbb{Q}(\sqrt{-1})\otimes_\mathbb{Q} \mathbb{Q}_3 $$ where $\mathbb{Q}_3$ denotes de $3$-adic rational numbers. Then $L$ is a quadratic extension of the local field $\mathbb{Q}_3$. Furthermore, the valuation ring of $L$ is $B:=\mathbb{Z}[\sqrt{-1}] \otimes \mathbb{Z}_3$.

It is known that the norm map $N$ maps the set $U_L$ of units in $B$ onto the set $U_{\mathbb{Q}_3}$ of units in $\mathbb{Z}_3$ (see Serre's book "Local Fields", Chapter V, Prop. 3). This implies that there is an element $x\in U_L$ such that $N(x)=-1$ since $-1$ is a unit in $\mathbb{Q}_3$. Could someone specify this element $x$?

oddprime and $u,v\in\mathbf{Z}_p^\times$, then $u$ is in the image of the norm map from $\mathbf{Q}_p(\sqrt{v})^\times$ down to $\mathbf{Q}_p^\times$. See for examples Serre'sCourse in arithmetic, Chapter III. $\endgroup$ – Chandan Singh Dalawat Dec 30 '11 at 6:44nt.number-theory. $\endgroup$ – Chandan Singh Dalawat Dec 30 '11 at 6:53