  Let $E$ be a field, $R$ a finite separable extension of $E$, 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 in each of
the following two special cases: (i) $E$ is primarily quasilocal and $R$ is an
intermediate field of a finite Galois extension $M/E$ with a nilpotent Galois
group; (ii) $E$ is quasilocal and the natural Brauer group homomorphism Br$(E)
\to $ Br$(L)$ is surjective, for every finite extension $L$ of $E$. It is also
used for describing the norm groups of formally real quasilocal fields, and of
Henselian discrete valued fields whose finite extensions are strictly PQL. The
paper proves that the condition on $G(M/E)$ in (i) cannot be weakened, and the
surjectivity of the homomorphism Br$(E) \to $ Br$(L)$ is essential for the
validity of (ii).
