# Structure of the units of the valuation ring of a finite extension of $\mathbb{Q}_p$

Let $$K$$ be a finite extension of the field of $$p$$-adic numbers $$\mathbb{Q}_p$$, call $$\mathcal{O}_K$$ the valuation ring of $$K$$ (i.e. the set of elements of $$K$$ with valuation greater or equal than zero). Call $$\mathcal{O}_K^\times$$ the multiplicative group of units of $$\mathcal{O}_K$$ (i.e. the group of elements of $$K$$ with zero valuation).

I have read that there is a group isomorphism $$\begin{equation*} \mathcal{O}_K^\times\cong \mathbb{Z}_p^{a}\times U \end{equation*}$$ for some $$a\in\mathbb{N}$$ and $$U$$ finite group. Could you tell me why, or where I can find a proof of this result?

I know that $$\mathcal{O}_K\cong \mathbb{Z}_p^{[K:\mathbb{Q}_p]}$$ as $$\mathbb{Z}_p$$-modules, but this isomorphism only involves the additive structure of $$\mathcal{O}_K$$.

This should be in any algebraic number theory textbook that mentions the $$p$$-adics e.g. Neukirch. Here is a rough sketch; the technical details on the $$p$$-adic logarithm are in Neukirch and other places.
Let $$\pi$$ be a uniformizer. One knows that the higher unit groups, $$U^{(n)} = 1 + (\pi)^n$$, are of finite index in $$\mathcal O_K^\times$$ and that the logarithm gives a homomorphism from $$U^{(n)}$$ under multiplication to $$(\pi)^n$$ under addition as $$\mathbb Z_p$$-modules. For sufficiently large $$n$$, this is an isomorphism. Since $$(\pi)^n$$ is isomorphic to $$\mathcal O_K$$ (under addition), we can see that it is a free $$\mathbb Z_p$$-module of rank equal to the degree of $$K$$ over $$\mathbb Q_p$$. Therefore, $$\mathcal O_K^\times$$ is a $$\mathbb Z_p$$-module of rank $$[K:\mathbb Q_p]$$ (using the finite index fact). So we can decompose it into a free part of that rank times some torsion; the torsion of $$\mathcal O_K^\times$$ is clearly given by the roots of unity.
Putting it all together, we obtain a slightly more precise version of your claim: $$\mathcal O_K^\times \cong \mathbb Z_p^{[K:\mathbb Q_p]} \times \{\textrm{roots of unity in K}\}$$