Solution
= Solution
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> isomorphism
$$
1+\mathfrak m_K^r\xrightarrow{\ \log\ }(\mathfrak m_K^r,+).
$$
Multiplication 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.