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The discrete valuation gives a split exact sequencesplit by $n\mapsto {\pi }^n$. Reduction gives a second exact sequenceThe 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

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

For a finite extension $K/{\mathbb{Q}}_p$, its square-class group of a field is $K^{\times }/(K^{\times }{)}^2$. When $p$ is odd, parity of the valuation and the square class of the unit residue give $C_2\times C_2$. Representatives are $1,u,\pi ,u\pi$, where $\pi$ is a uniformizer of $K$ and $u$ has nonsquare residue. The Hensel lemma lifts square roots of units with square residue.

For $K={\mathbb{Q}}_2$, the group is $C_2^3$, with generators $-1,5,2$; a unit is a square exactly when it is one modulo eight. More generally, if $K/{\mathbb{Q}}_2$ has degree $d$, the group has order $2^{d+2}$. Indeed, the principal-unit group of a mixed-characteristic local field has a finite cyclic 2-power torsion factor and a free ${\mathbb{Z}}_2$ factor of rank $d$, each contributing its quotient by squares, and the valuation contributes one further factor $C_2$. The specific representatives $1,u,p,up$ therefore apply to ${\mathbb{Q}}_p$ for odd $p$; a ramified extension needs its own uniformizer.