This is related to Iwasawa's Local Class Field Theory Chpt 3 and 4.
Let $k$ be a local field and $K$ be maximal unramified algebraic extension of $k$. Set $\Omega$ the algebraic closure of $k$. Take $\bar{K}$ and $\bar{\Omega}$ as completed fields under extension of discrete valuation from $k$. Suppose $F$ is a discrete valued field. Denote $O_F$ as the valuation ring associated to $F$. Let $F_q$ be the residue field.
Set $q$ cardinality of residue field of $k$. Let $f\in O_{\bar{K}}[[x]]$ s.t. $f-\pi x\in (x)^2$ and $f=x^q$ under reduction against prime ideal of $O_{\bar{K}}$ where $\pi$ is any prime ideal generator of $O_{\bar{K}}$.
Then it follows from the book that there is a unique formal group law $F_f(X,Y)$ associated to $f$ s.t. $f\in Hom(F_f,F_f^\phi)$ where $\phi$ is frobenius automorphism identified with $Gal(\bar{K},k)$ due to unramified extension... etc
$\textbf{Q:}$ Why does this $f$ work and magically giving rise to formal group law?(One of possible reason is that this $f$ allows pinning down various power series $F_f$ uniquely but this is looking from back. The proof is not too hard but it is not short or obvious at first sight. Roughly, there is a unique $F$ s.t. $F(X_1,\dots, X_n)$ s.t. $f\circ F=F^\phi\circ f$.) In other words, why should I think this $f$ is a "godly" given choice in that particular form?(Why I can't take $f-\pi x^2\in (x)^3$ and $f=x^{q^2}$ instead? It is possible to twist $f$ by invertible power series $\theta$. Consider $f^\theta=\theta\circ f\circ \theta^{-1}$ where $\theta\in O_{\bar{K}}[[x]]$. It is easy to check if $f$ has property boxed above, then $f^\theta$ will also have the same property.) It is clear that $f$ contains both prime element information by leading term and Frobenius map information $x^q$.