= Multiplicative group of a non-Archimedean local field
{title2=$K^\times$}
If a non-Archimedean <local field> $K$ has <uniformizer> $\pi$, <valuation ring> $\mathcal O_K$, maximal ideal $\mathfrak m_K$ and residue field $\mathbb F_q$, then
$$
K^\times\cong\pi^{\mathbb Z}\times\mu_{q-1}\times(1+\mathfrak m_K).
$$
The first factor records the <discrete valuation>, the <Teichmuller representatives> split reduction $\mathcal O_K^\times\to\mathbb F_q^\times$, and $1+\mathfrak m_K$ is the group of <principal units>.
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