Norm subgroup of a local field extension ($N_{L/K}(L^{\times })$)

For a finite extension of local fields $L/K$, its norm subgroup is the image of the field norm $N_{L/K}:L^{\times }\to K^{\times }$. For an abelian extension, it is the kernel of the induced local Artin map to $\mathrm{Gal}(L/K)$.

For a finite unramified extension of non-Archimedean local fields, the field norm on residue units is surjective: on a cyclic group of order $q^n-1$ it is exponentiation by $(q^n-1)/(q-1)$ and has image of order $q-1$. For a common uniformizer $\pi$ and $r\ge 1$, expansion of the conjugate product givesThe field trace of a finite finite field extension is onto: its polynomial $X+X^q+\cdots +X^{q^{n-1}}$ is nonzero of degree below $q^n$, so defines a nonzero linear map to the base field. Lift a residue norm and correct successively with elements of $1+{\pi }^r{\mathcal{O}}_L$. The infinite product converges by completeness, and its norm is the prescribed unit. This argument also works in positive characteristic and when $p$ divides $n$.

For a cyclic extension of p-adic fields, the Herbrand quotient of the local multiplicative group together with Hilbert theorem 90 gives this index. If the ramification index and residue degree are $e,f$, the norm valuation formula $v_K(Nx)=f{v}_L(x)$ then gives the unit norm index $e$. In particular all units are norms in an unramified extension.

If $L/K$ is a finite abelian extension of local fields, thenIndeed, Local Artin reciprocity gives total norm index $[L:K]=ef$, while the valuation image of the norm subgroup is $f\mathbb{Z}$.

For a tamely totally ramified extension of degree $q-1$ of a local field with residue field ${\mathbb{F}}_q$, the unit norms are exactly the principal units. Reduction sends a unit norm to its $(q-1)$st power, and the norm-unit index in an abelian local extension shows that this containment has the correct index.

For every positive integer $n$,Under Local Artin reciprocity, the cyclotomic character identifies the quotient with $(\mathbb{Z}/2^n\mathbb{Z}{)}^{\times }$: a uniformizer acts trivially on the totally ramified extension, while a 2-adic unit acts through its residue class modulo $2^n$.