On the equality of norm groups of global fields

Abstract

Let k be a global field. In an earlier work we proved that K ⊆ L iff N L k L ∗ ⊆ 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 ∗ ⊆ 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 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 ≤ 5. We also show that there exists a pair of nonequivalent norm forms of degree 6 with equal value sets.