Is the Field Norm of a unit a unit?

Given a finite extension $K/L$ and a subring $A \subset K, Frac(A)=K$, $B = A \cap L$,

For $a \in A$, let $R_1 = B[a]$, and pick iteratively some $a_{m+1} \in A$ such that $R_{m+1}= R_m[a_{m+1}]$ is a free $R_m$-module, until $R_M$ is of finite index in $A$. So $R_M$ is also a free $B$-module.

The field norm $N_{K/L}(a)$ is the determinant of the $B$-linear map $x \mapsto a x , R_M \to R_M$. So $N_{K/L}(a) \in B$.

If $a \in A^\times$ then $1=N_{K/L}(aa^{-1})=N_{K/L}(a)N_{K/L}(a^{-1})$ and hence $N_{K/L}(a) \in B^\times$.