Let $K \mid \mathbb{Q}_2$ be a finite unramified extension of fields of degree $f$, i.e., $2 \in \mathbb{Z}_2$ is a uniformizer in $\mathcal{O}_K$ and $\mathcal{O}_K/2 \mathcal{O}_K \cong \mathbb{F}_{2^f}$ is a field containing precisely $2^f$ elements.
What can be said about the structure of the unit groups $K^\times$ and $\mathcal{O}_K^\times$?
It is clear that $K^\times \cong \mathbb{Z} \times \mathcal{O}_K^\times$. Further, $\exp$ and $\log$ induce an isomorphism $$(\mathcal{O}_K,+) \cong (4 \mathcal{O}_K,+) \cong (1+4\mathcal{O}_K, \cdot) .$$ Now I would like to write $$\mathcal{O}_K^\times \cong G \times (1+4\mathcal{O}_K)$$ for some group $G$. It should be $$|G| = |\mathcal{O}_K/4 \mathcal{O}_K| = |(\mathcal{O}_K/2 \mathcal{O}_K)^\times| \cdot |\mathcal{O}_K/2 \mathcal{O}_K| = (2^f-1) \cdot 2^f.$$ The ''$(2^f-1)$-part'' of $G$ could be the subgroup $\mu_{2^f-1}(K)$ of $(2^f-1)$-th roots of unity in $K$.
My question now is whether such a group $G$ actually exists and how to construct the isomorphism $$\mathcal{O}_K^\times \cong G \times (1+4\mathcal{O}_K).$$ Does anyone have an idea?