On the scope of validity of the norm limitation theorem for quasilocal fields

Let $E$ be a quasilocal field, $R/E$ a finite separable extension, and $R _{\rm ab}$ the maximal abelian subextension of $E$ in $R$. The main result of this paper shows that the norm groups $N(R/E)$ and $N(R_{\rm ab}/E)$ are equal, if the natural Brauer group homomorphism Br$(E) \to$ Br$(L)$ is surjective, for every finite extension $L$ of $E$. The paper proves in a strong form that the surjectivity of the homomorphisms Br$(E) \to$ Br$(L)$ is essential.