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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

Let ${\mathfrak{m}}_K$ be the maximal ideal. For all sufficiently large $r$, the convergent p-adic logarithm and p-adic exponential give the principal-unit logarithm isomorphismMultiplication by a power of a uniformizer identifies the additive group ${\mathfrak{m}}_K^r$ with $({\mathcal{O}}_K,+)$. Since $1+{\mathfrak{m}}_K^r$ has finite index in ${\mathcal{O}}_K^{\times }$, it is the required subgroup.

The smallest integer isIndeed, reduction modulo $p$ cannot work: the map $x\mapsto x^p$ is the identity on ${\mathbb{F}}_p^{\times }$, although not every element of ${\mathbb{Z}}_p^{\times }$ is a $p$th power.

For odd $p$, the p-adic unit group decomposes asRaising to the $p$th power is an automorphism on ${\mu }_{p-1}$. On the principal units, the p-adic logarithm identifies it with multiplication by $p$ on $p{\mathbb{Z}}_p$, soHence whether a unit is a $p$th power is determined exactly by its residue modulo $p^2$. Equivalently,This is the pth-power criterion for p-adic units.

For sufficiently large $r$, the series for the p-adic logarithm and p-adic exponential converge on $1+{\pi }^r{\mathcal{O}}_K$ and ${\pi }^r{\mathcal{O}}_K$ and are inverse homomorphisms. Hence the principal-unit logarithm giveswhere the last map is division by ${\pi }^r$.

Now take $K={\mathbb{Q}}_3({\zeta }_3)$ and $\pi ={\zeta }_3-1$. This is a uniformizer, the residue field is ${\mathbb{F}}_3$, and its Teichmuller units are ${\mu }_2=\{\pm 1\}$. ThusThe image of ${\zeta }_3$ generates $U_1/U_2$, and ${\mu }_3\cap U_2=1$, so $U_1={\mu }_3\times U_2$. The logarithm and exponential already converge inversely on $U_2$, giving $U_2\cong ({\pi }^2{\mathcal{O}}_K,+)\cong ({\mathcal{O}}_K,+)$. Since ${\mu }_2\times {\mu }_3\cong {\mu }_6$, this proves the unit group of Q3 zeta3 decomposition

Suppose first that $x^p+y^p=z^p$ with $x,y,z\in {\mathbb{Z}}_p^{\times }$. Divide by $y^p$ and put $u=x/y$, $w=z/y$. Reduction modulo $p$ gives $w\equiv u+1(\mathrm{mod}p)$. Choose an integer $a\equiv u(\mathrm{mod}p)$; then $p\nmid a(a+1)$. A unit's $p$th power modulo $p^2$ depends only on its residue modulo $p$, because $(a+pt{)}^p\equiv a^p(\mathrm{mod}{p}^2)$. Therefore

Conversely, suppose such an integer $a$ exists. The congruence saysFor odd $p$, the $p$th-power map sends $1+p{\mathbb{Z}}_p$ onto $1+p^2{\mathbb{Z}}_p$: under the p-adic logarithm it becomes multiplication by $p$. Hence the displayed ratio is $t^p$ for some $t\in 1+p{\mathbb{Z}}_p$. Takingproduces the required solution in ${\mathbb{Z}}_p^{\times }$.

For sufficiently large $r$, the convergent p-adic logarithm and p-adic exponential series are inverse homomorphismsTheir identities $\log (xy)=\log x+\log y$ and $\exp (x+y)=\exp x\exp y$ follow first formally and then by convergence. Multiplication by ${\pi }^r$ identifies $({\mathcal{O}}_K,+)$ with $({\pi }^r{\mathcal{O}}_K,+)$. Since $(1+{\pi }^r{\mathcal{O}}_K)$ has finite index in ${\mathcal{O}}_K^{\ast }$, the conclusion follows.

For a finite extension $K/{\mathbb{Q}}_p$ and all sufficiently large $r$, the p-adic logarithm and p-adic exponential are inverse group isomorphisms

Choose ordered minimal topological generators $a_1,\dots ,a_d$ of a uniform pro-p group for odd $p$, put $b_i=a_i-1$, and choose $1/(p-1)<s<1$. Ordered monomials $b^{\alpha }=b_1^{{\alpha }_1}\cdots b_d^{{\alpha }_d}$ give power-series coordinates in the integral completed group algebra. The displayed Non-Archimedean absolute value filtration gives a complete normed ${\mathbb{Q}}_p$-algebra after completion of the rational group algebra. The power relation raises valuation by one, and interchanging $b_i,b_j$ adds terms of valuation at least $1+s>2s$, so multiplication does not decrease total valuation. The p-adic logarithm and p-adic exponential are inverse on the relevant small balls. In particular $\log G$ is a faithful realization of the Lie algebra of a uniform pro-p group, useful for verifying its intrinsic limit operations and the Baker--Campbell--Hausdorff formula without assuming the correspondence in advance.