For sufficiently large $r$, the convergent <p-adic logarithm> and <p-adic exponential> series are inverse homomorphisms
$$
\log:1+\pi^r\mathcal O_K\longleftrightarrow\pi^r\mathcal O_K:\exp.
$$
Their 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 $\mathcal O_K^*$, the conclusion follows.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 136 / 3 / a / Solution

{title2=$\exp_p$}
{wiki=P-adic_exponential_function}

The $p$-adic exponential $\exp(x)=\sum_{n\geq0}x^n/n!$ converges near zero and is locally inverse to the $p$-adic logarithm.


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