TITLE: Ring of integers and unit group for local fields
QUESTION [5 upvotes]: On page 6 of Sutherland's notes (https://math.mit.edu/classes/18.785/2018fa/LectureNotes27.pdf), he claims that 
$$\mathcal{O}_K^\times\cong\mathbb{F}_p^\times\times\mathcal{O}_K.$$I understand the structure theorem (from Neukirch, for instance) tells us that $$\mathcal{O}_K^\times\cong\mathbb{F}_p^\times\times U^{(1)}.$$ In both cases, these are isomorphisms of topological groups. I don't quite see why $\mathcal{O}_K\cong U^{(1)}$ -- does it follow from some application of the p-adic logarithm (note that the first is an additive group and the latter is a multiplicative group).

REPLY [4 votes]: Even though you certainly didn’t ask about the case of characteristic $p$, the excellent answer of @reuns begs for the extension below.
The case of a finite extension of $\Bbb F_p((x))$ is nothing like the situation in characteristic zero. First, the only locally compact fields in positive characteristic are the $\Bbb F_q((x))$, $q$ being a power of $p$. That’s the easy part. Also, the units are still $\Bbb F_q^\times\times U^{(1)}$; I’ll write the second factor as $1+\mathfrak m$. But this latter is torsion-free, and thus a torsion-free $\Bbb Z_p$-module. But not finitely generated, as you’ll see.
A moment’s reflection tells you that the series $\{1+x^m\}$ for $p\nmid m$ all are $\Bbb Z_p$-independent, multiplicatively of course. And in the simplest case $q=p$, one gets
$$1+\mathfrak m\cong\prod_{p\nmid m}\Bbb Z_p\,,$$ that is, a product (not direct sum) of those infinitely many copies of $\Bbb Z_p$.
For the general case $q>p$, the story is more complicated, best illuminated by means of the dreaded Artin-Hasse Exponential, which is rather hard to understand and to use. The upshot, though, is that in the general case, you get
$$
1+\mathfrak m\cong\prod_{p\nmid m}W_\infty(\Bbb F_q)\,,
$$
those $W\!$s being the Witt rings, each being isomorphic to the integers in the unramified extension of $\Bbb Q_p$ of degree $[\Bbb F_q:\Bbb F_p]$.