Unit group of power series ring Is there any way to calculate the multiplicative group of the units of power series ring $k[[x]]$, where $k$ is a field ?
 A: Hint $\rm\displaystyle\quad 1\: =\: (a-xf)(b-xg)\ \Rightarrow\ \color{#c00}{ab=\bf 1}\ $ so scaling top & bottom below by $\rm \,b\,$ yields
$$\Rightarrow\ \ \displaystyle\rm\ \ \frac{1}{a-xf}\ =\ \frac{b}{\color{#c00}{\bf 1}-bxf}\ =\ b\:(1+bxf+(bxf)^2+(bxf)^3+\:\cdots\:)$$
A: $\bf Hint:$ $\sum_{n=0}^\infty a_nx^n$ is a unit iff $a_0\ne 0$.
A: The multiplicative group is $k[[x]]\backslash (x)$.
Certainly those elements divisible by $x$ are not units. If an element is not divisble by $x$ (in other words, has nonzero constant term), you can construct the inverse term by term.
A: It's possible to be much more specific about the structure of the unit group than has been done so far, as follows. So far we know that the units in $k[[x]]$ are those formal power series with nonzero constant term. Dividing by the constant term shows that this group is isomorphic to $k^{\times}$ (the constant term part) times the group of formal power series $1 + a_1 x + a_2 x^2 + \dots$ with constant term $1$.

Claim: If $k$ has characteristic $0$, then this group is isomorphic to $(k[[x]], +)$. In particular, it is a $\mathbb{Q}$-vector space. 

(Note that this can't be true in characteristic $p$ since the group of units is not $p$-torsion: we have, for example, $(1 + x)^p = 1 + x^p \neq 1$.) 
The isomorphism is given by the "exponential" map
$$k[[x]] \ni f \mapsto (1 + x)^f \in k[[x]]^{\times}.$$
This is defined using the identity
$$1 + x = \exp \log (1 + x) = \exp \sum_{n \ge 1} (-1)^{n-1} \frac{x^n}{n}$$
(which is where we need that $k$ has characteristic $0$), which allows us to define
$$(1 + x)^f = \exp f \log (1 + x)$$
as a formal power series. This has all of the standard properties of the exponential, and is in particular a homomorphism. The inverse of this isomorphism is the "logarithm" map
$$\log_{1 + x}(g) = \frac{\log g}{\log (1 + x)}$$
where $g$ has constant term $1$. It's formal to verify that these two operations are inverse to each other. 
The answer is more interesting in positive characteristic: we get the underlying abelian group of the ring of Witt vectors $W(k)$. For example, when $k = \mathbb{F}_p$ this group is related to the group of $p$-adic integers $\mathbb{Z}_p$, and in fact there is a natural exponential $(1 + x)^f \in \mathbb{F}_p[[x]]$ where $f \in \mathbb{Z}_p$. Edit: See the comments!
