= Solution
The <discrete valuation> gives a split exact sequence
$$
1\longrightarrow\mathcal O^\times\longrightarrow K^\times
\xrightarrow{v_K}\mathbb Z\longrightarrow0,
$$
split by $n\mapsto\pi^n$. Reduction gives a second exact sequence
$$
1\longrightarrow1+\mathfrak m\longrightarrow\mathcal O^\times
\longrightarrow\mathbb F_q^\times\longrightarrow1.
$$
The <Teichmuller lift> identifies the cyclic group $\mathbb F_q^\times$ with the group $\mu_{q-1}$ of prime-to-$p$ <roots of unity> in $K$, where $p$ is the <residue characteristic>. Consequently the <multiplicative group of a non-Archimedean local field> decomposes as
$$
\boxed{K^\times\cong\mathbb Z\times\mu_{q-1}\times(1+\mathfrak m).}
$$
If $d=[K:\mathbb Q_p]$, the <p-adic logarithm> identifies a sufficiently deep <higher principal-unit group> with the additive group of a rank-$d$ $\mathbb Z_p$-lattice. The remaining kernel is precisely the finite group $\mu_{p^a}(K)$ of $p$-power roots of unity in $K$. Thus the <principal-unit group of a mixed-characteristic local field> has the topological group structure
$$
\boxed{1+\mathfrak m\cong\mu_{p^a}(K)\times\mathbb Z_p^d.}
$$
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