Summary: Let k be a global field. In an earlier work [ibid. 32, 203-219 (1989; Zbl 0687.12006)] we proved that \(K\subseteq L\) iff \(N_{L/k}L^*\subseteq N_{K/k}K^*\) for any finite Galois extensions K, L of k. In this paper we investigate the equality of norm groups corresponding to finite separable extensions of k. Let K, L be extensions of k contained in a finite Galois extension E of k. We prove that \(N_{K/k}K^*\) is almost contained in \(N_{L/k}L^*\) (i.e., the intersection of the norm groups is a subgroup of finite index in \(N_{K/k}K^*)\) iff every element of G(E/K) of prime power order is a conjugate of an element of G(E/L) in G(E/k). This criterion yields a number of interesting corollaries; some of these are the following. Every norm group is a subgroup of infinite index in \(k^*\). If \(N_{K/k}K^*\subseteq N_{L/k}L^*\), then the normal core of L/k is contained in the normal core of K/k. Furthermore, if K, L are contained in a finite nilpotent extension of k, then \(N_{K/k}K^*=N_{L/k}L^*\) implies that K, L have the same normal closure over k. We also show that for any cubic Galois extension L/k there exist an infinite number of quadratic extensions K of L with \(N_{K/k}K^*=N_{L/k}L^*.\)
In an earlier paper (loc. cit.) we proved that two norm forms associated with two Galois extensions of k are equivalent (over k) iff they have the same value set. In this paper we prove that this result is true for any pair of norm forms f, g with deg f\(=\deg g\leq 5\). We also show that there exists a pair of nonequivalent norm forms of degree 6 with equal value sets.