Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 136 / 4 / b

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