Norm groups of global fields (Q1345287)

\textit{L. Stern} [J. Number Theory 32, 203-219 (1989; Zbl 0687.12006); ibid. 36, 108-126 (1990; Zbl 0718.11056)] has proved the following theorems: (i) Let \(K/k\) and \(L/k\) be finite Galois extensions of a global field \(k\). Then \(K=L\) if and only if \(N_{K/k} K^ \times= N_{L/k} L^ \times\). (ii) If \(K/k\) is a nontrivial finite separable extension of global fields, then \(k^ \times/ N_{K/k} K^ \times\) is infinite. In this note, the author shows that these two results follow easily from the Chebotarev density theorem. The method of proof is to state the results for the images of the norm map on divisors rather than the element norm.