Reference request for the group of units of a power series ring in one variable

Question

Let $\mathbb F_p$ denote the finite prime field of $p$ elements. What reference can be recommended for an analysis of the structure of the group of units of the power series ring $\mathbb F_p[[x]]$? There must be much known more generally when the finite field is replaced by any other field but I am most interested in the finite prime case.

Answer 1

In general the multiplicative group $1+x\,A[\![x]\!]\leq U(A[\![x]\!])$ is isomorphic to the additive group of the ring $W(A)$ of big Witt vectors of $A$. If $A$ is $p$-local for some prime $p$, then $W(A)$ is additively isomorphic to a product of copies of the ring $W_{p^\infty}(A)$ of $p$-typical Witt vectors, with one copy for each positive integer that is coprime to $p$. If $A$ is a finite field of order $p^d$, then $W_{p^\infty}(A)$ is a complete discrete valuation ring that is additively isomorphic to $\mathbb{Z}_p^d$ and has a natural ring isomorphism $W_{p^\infty}(A)/p\simeq A$; moreover, $W_{p^\infty}(A)$ is uniquely characterised by these properties, up to canonical isomorphism. More concretely, $A$ can be described as $\mathbb{F}_p[t]/f(t)$ for some polynomial $f(t)$ over $\mathbb{F}_p$ that divides the cyclotomic polynomial $\varphi_{p^d-1}(t)$, and there is a unique lift of $f(t)$ to $\mathbb{Z}_p[t]$ that still divides the cyclotomic polynomial, and we have $W_{p^\infty}(A)=\mathbb{Z}_p[t]/f(t)$.

For more details you can read these notes by Lars Hesselholt, or the references listed there.