On the scope of validity of the norm limitation theorem in one-dimensional abstract local class field theory

Abstract

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).