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Let $K$ be a local field and $L/K$ be a finite extension. Let $L^{ab}$ be the maximal abelian subextension of $K$ in $L$. Write $N_L$ (resp. $N_{L^{ab}}$) for the image of the norm map from $L$ (resp. $L^{ab}$) to $K$; this is a subgroup of $K^\times$.

Two standard consequences of local class field theory are:

  • $N_L$ is open and of finite index in $K^\times$.
  • (Norm limitation theorem) $N_L=N_{L^{ab}}$.

All of this is explained, for instance, in Milne's notes on class field theory, section III.3.

My question is: are there proofs of these facts which do not use local class field theory?

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    $\begingroup$ In Fesenko's book on local fields, chapter 3 and 4, there are elementary computations of the norm maps and groups. $\endgroup$
    – user19475
    Commented Jan 16, 2019 at 13:27
  • $\begingroup$ @TKe Thank you for this reference. Do you mean the book "Local fields and their extensions" by Fesenko and Vostokov? As far as I can tell from a quick read of those chapters, the results I mention on norm subgroups are still deduced from the existence of the local Artin reciprocity map. Their approach to this local Artin map is "elementary" in the sense that it does not appeal to either group cohomology or central simple algebras. I am looking for an approach which proves the results in a qualitative fashion, without pinning down the precise relation with Galois groups. $\endgroup$
    – Simon Pepin Lehalleur
    Commented Jan 17, 2019 at 7:58
  • $\begingroup$ I think till (1.5) of Chapter 4, no reciprocity map is involved (but some Galois theory), and these results should give the finite index of the norm group, hence open if the index is not divisible by the residue characteristic. $\endgroup$
    – user19475
    Commented Jan 17, 2019 at 8:21

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