Edit: For answers read the comments at this address: https://mathoverflow.net/questions/424905/about-lubin-tate-extensions
This is from the $6^{th}$ section of the last chapter of the book class field theory by Childress:
(Page 201): "We shall use the polynomials that give rise to Lubin-Tate formal group laws to construct certain totally ramified abelian extensions of a local field $K$. Let $q =$ #$\mathbb{F}_K$ and, as before, let $\mathcal{F}_{\pi_K}$ denote the set of all power series $f (X) ∈ \mathcal{O}_K [[X]]$ such that $f (X) \cong \pi _K X (\mod X^2)$ and $f (X) \cong X^q (\mod \pi_K )$. Suppose $f(X)\in \mathcal{F}_{\pi_K}$ is a monic polynomial of degree $q$. Then $$f (X) = X^q + \pi_K (a_{q−1}X^{q−1} +· · ·+a_2X^2) + \pi_K X,$$ where $a_j ∈ \mathcal{O}_K$. For a positive integer $m$, define recursively $$f ^{(1)}(X) = f (X), f ^{(2)}(X) = f ( f (X)), . . . , f ^{(m)}(X) = f ( f ^{(m−1)}(X)).$$ By (i ) and (ii ) of Corollary 5.3, we have $$f (X) = [\pi_K ]_f (X), \space\space \space and \space f ^{(m)}(X) = [\pi_K^m ]_f (X).$$ It follows that if $\lambda \in \mathcal{P}_f$ , (the Lubin-Tate module), then $$f ^{(m)}(X) = \pi_K^m \space ·_f \lambda."$$
(The major question) Look at the bold lines above. I can not see how an arbitrary $\lambda$ played this role? This $\lambda$ seems to be chosen arbitrarily. Does it have a typo? Should it written as "$f ^{(m)}(\lambda) = \pi_K^m \space ·_f \lambda$"? Then I think everything is correct and compatible.
(Page 200) About corollary 5.3 mentioned above: From the corollary 5.3, the power series $[a]_f (X)$ allows us to define a formal $\mathcal{O}_K$ -module structure when combined with the formal group law $F_f$. And after that, the book defined two operations $+_f$ and $._f$ to make $\mathcal{P}_{\Omega}$ an $\mathcal{O}_K$-module (which the author denote $\mathcal{P}_f$). Especially the operation $._f$ defined as follow:
- For $a \in \mathcal{O}_K$ and $x\in \mathcal{P}_{\Omega}$, the series $[a]_f (x)$ converges to an element of $\mathcal{P}_{\Omega}$, which we denote $a \space ·_f x$.
(Pages 182 & 183) Some of notations defined at the beginning of the chapter:
- $\mathcal{P}_K = \{x \in K : v_K (x) > 0\} = \pi_K\mathcal{O}_K$,
- $\Omega$ a complete, algebraically closed extension of $\hat{K}_{ur}$
(The minor question on the definition of $\mathcal{P}_{\Omega}$) It seems the author forgot to define $\mathcal{P}_{\Omega}$, but I think it should be something very similar to $\mathcal{P}_K$. But I can not write its definition, because I think we don't have a uniformizer in $\Omega$. How can we define a valution on $\Omega$ in the absence of uniformizers?