= Uniformizer character of inertia
{title2=$\theta_0(\sigma)=\overline{\sigma(\pi_L)/\pi_L}$}
For a finite Galois extension of non-Archimedean <local fields>, this is a homomorphism from the <inertia group> to $k_L^\times$ with kernel the <wild inertia group>. Inertia acts trivially on residue units, which proves multiplicativity and independence of the chosen <uniformizer>. Passing to the maximal unramified subextension reduces the kernel calculation to a totally ramified extension. Hence it embeds the <tame inertia quotient> into the cyclic multiplicative group of the residue field.
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