# Unit group and local class field theory

Let $k$ be a local field, and $U$ the group of units.

"by local class field theory, there is a continuous surjective homomorphism $U \to I$"

where $I$ is the intertia groups at a certain prime of $k$.

Can someone point me to the result in local class field theory from which this follows and/or give a brief derivation if it isn't obvious? I have Neukirch's book on ANT at hand, and I can get Lang's.

• Neukirch, Artin map in the local case. – franz lemmermeyer Aug 1 '16 at 9:30

As an overview, for an abelian extension of local fields $K^{\times}$ surjects onto $Gal(L/K)$, with kernel equal to the norm group. Under this map, $\mathcal{O}_K^{\times}$ corresponds to inertia, with the torsion elements (ie roots of unity) corresponding to the tame part.
• Your comment about torsion is not quite correct. It is the prime-to-$p$ roots of $1$, i.e. the Teichmuller lifts of the units in the residue field, that correspond to tame inertia. E.g. if $K = \mathbb Q_2$, then $\mathbb Z_2^{\times}$ contains $-1$ as a $2$-torsion element, which corresponds to a degree 2, and thus wildly ramified, extension. – tracing Aug 5 '16 at 3:59