Norm Groups of Global Fields

AbstractLeonid Stern (1989, J. Number Theory32, 203-219; 1990, J. Number Theory36, 127-132) proves that two finite Galois extensions of a global field are equal if the images of the norm maps are equal and that, for a nontrivial finite separable extension of global fields, the image of the norm has infinite index. In this note we show that these results follow easily from Tchebotarev density. We do this by first proving the results for the images of the norm map on divisors and then by showing that if the images of the norm maps of two extensions are almost equal then the corresponding images of the divisor norm maps are almost equal also