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

In a proof I read

"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.

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    $\begingroup$ Neukirch, Artin map in the local case. $\endgroup$
    – franz lemmermeyer
    Aug 1, 2016 at 9:30

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What you need is the local Artin map. This is briefly mentioned in Neukirch in the Local Class Field Theory chapter, but its construction is nontrivial.

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.

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  • $\begingroup$ 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. $\endgroup$
    – tracing
    Aug 5, 2016 at 3:59

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