Let $K$ be a field, $L/K$ be a finite extension, $M/L$ be an infinite algebraic extension, must we have $$\bigcap_{L\subset L'\subset M,[L':L]<\infty} \mathrm{Nm}_{L'/K}(L'^\times) = \mathrm{Nm}_{L/K}\left(\bigcap_{L\subset L'\subset M,[L':L]<\infty} \mathrm{Nm}_{L'/L}(L'^\times)\right),$$ where $\mathrm{Nm}_{L'/K}$ is the field norm, and $\mathrm{Nm}_{L'/K}(L'^\times) = \{\mathrm{Nm}_{L'/K}(x):x\in L'\setminus\{0\}\}$?
The LHS is $\displaystyle{\bigcap_{L\subset L'\subset M,[L':L]<\infty}} \mathrm{Nm}_{L/K}(\mathrm{Nm}_{L'/L}(L'^\times))$, so obviously one have RHS $\subset$ LHS. $[M:L]$ is supposed to be infinite, otherwise both sides are trivially equal to $\mathrm{Nm}_{M/K}(M^\times)$.