= Solution
Fix a non-Archimedean <local field> $K$, its <valuation ring> $R=\mathcal O_K$, a <uniformizer> $\pi$, and a <residue field> with cardinality $q$. The construction below applies both in mixed and equal characteristic. A one-dimensional commutative <formal group law> is $F(X,Y)\in R[[X,Y]]$ with identity, associativity and commutativity, and $F(X,Y)=X+Y+$ higher-degree terms. It has a unique <formal inverse>. Its morphisms are series $h(T)\in TR[[T]]$ with $h(F(X,Y))=G(h(X),h(Y))$; a <unit> linear coefficient makes such a morphism invertible by recursive coefficient comparison. Evaluating its integral series on the <maximal ideal> of a finite extension of $K$ is legitimate because the series converge there.
Choose a <Lubin–Tate series> $f(T)$ satisfying
$$
f(T)=\pi T+O(T^2),\qquad f(T)\equiv T^q\pmod\pi.
$$
A useful choice is $f(T)=\pi T+T^q$. The <Lubin–Tate functional equation lemma> constructs a unique series with any prescribed linear form $\ell$ and satisfying
$$
f(H(X_1,\ldots,X_r))=H(f(X_1),\ldots,f(X_r)).
$$
Here is its coefficient argument. If terms below degree $d$ have been found, the unknown homogeneous degree-$d$ term appears with multiplier $\pi-\pi^d$. The already known error has every coefficient divisible by $\pi$: after reduction, the equation becomes $H(X)^q=H(X_1^q,\ldots,X_r^q)$, valid because the residue coefficients lie in $\mathbb F_q$. Since $\pi-\pi^d$ is $\pi$ times a <unit> for $d\geq2$, the next term is uniquely determined and integral. Recursion proves existence and uniqueness.
Use $\ell=X+Y$ to construct $F_f$, and $\ell=aT$ to construct $[a]_f$ for every $a\in R$. Uniqueness gives $F_f(X,0)=X$ and $F_f(0,Y)=Y$. Comparing $[a]_f(F_f(X,Y))$ with $F_f([a]_f(X),[a]_f(Y))$, which satisfy the same functional equation and have the same linear form, proves that $[a]_f$ is an endomorphism of the law. The same uniqueness applied to three variables proves associativity: $F_f(F_f(X,Y),Z)$ and $F_f(X,F_f(Y,Z))$ have the same linear form and functional equation. It also proves commutativity and the identities
$$
[a+b]_f(T)=F_f([a]_f(T),[b]_f(T)),\qquad
[ab]_f=[a]_f\circ[b]_f,\qquad [1]_f(T)=T,\qquad [\pi]_f=f.
$$
Thus this <Lubin–Tate formal group> is equipped with an action of the whole ring $R$, not only its integers. For a <unit> $a$, the inverse of $[a]_f$ is $[a^{-1}]_f$. The same recursion, with two series for the fixed $\pi$, gives a unique strict <Lubin–Tate change of series>. Its integral series and inverse converge on torsion points, so it identifies their torsion fields. We may compute with $f(T)=\pi T+T^q$ without losing the general construction for that uniformizer.
Write $f_0(T)=T$ and $f_n=f^{\circ n}$. The <Lubin–Tate torsion> of level $n$ is
$$
\Lambda_n=\{x\text{ algebraic over }K:|x|<1,\ f_n(x)=0\}.
$$
For $n\geq1$, its primitive points are the roots of
$$
\Phi_n(T)=\frac{f_n(T)}{f_{n-1}(T)}=\pi+f_{n-1}(T)^{q-1}.
$$
This is a <monic> <polynomial> of degree $D_n=(q-1)q^{n-1}$. Modulo $\pi$ it is $T^{D_n}$, and its constant coefficient is exactly $\pi$. Thus the <Eisenstein layers of Lubin–Tate torsion> are <Eisenstein polynomials>. A root $\lambda_n$ generates a <totally ramified extension> of degree $D_n$, and has $v_K(\lambda_n)=1/D_n$ in the uniquely extended <valuation>. Also $f_{n-1}(\lambda_n)^{q-1}=-\pi\ne0$, so it has exact level $n$.
The map
$$
R/\pi^nR\longrightarrow\Lambda_n,\qquad a\longmapsto[a]_f(\lambda_n)
$$
is well-defined and injective. If $a=\pi^r u$ with $r<n$ and $u$ a <unit>, then $[a]\lambda_n=[u]f_r(\lambda_n)\ne0$, because $[u]$ is invertible and $\lambda_n$ is primitive. Conversely every multiple of $\pi^n$ kills it. The image therefore contains $q^n$ distinct roots of the degree-$q^n$ <polynomial> $f_n$, so it is all of $\Lambda_n$. This proves, without merely assuming the torsion cardinality,
$$
\boxed{\Lambda_n\cong R/\pi^nR\quad\text{as an }R\text{-module}.}
$$
Its primitive generators are exactly $[a]\lambda_n$ with $a\in(R/\pi^nR)^{\times}$.
Define the <Lubin–Tate extension> $K_{\pi,n}=K(\Lambda_n)$. Every integral endomorphism series evaluated at $\lambda_n$ converges in the complete field $K(\lambda_n)$, so all torsion points already lie there. Hence
$$
K_{\pi,n}=K(\lambda_n),\qquad [K_{\pi,n}:K]=D_n.
$$
It is the splitting field of $f_n$, which has $q^n$ distinct roots by the <module> calculation. Thus it is Galois. Every K-automorphism preserves the uniquely extended <absolute value on a field>, hence is continuous and commutes with limits of the integral endomorphism series. Its automorphisms therefore commute with the endomorphism action and send a primitive generator to a primitive generator. There is a unique <unit> class $a_\sigma$ such that $\sigma(\lambda_n)=[a_\sigma]\lambda_n$. The <Lubin–Tate Galois action on primitive torsion> gives an injective homomorphism
$$
\operatorname{Gal}(K_{\pi,n}/K)\longrightarrow(R/\pi^nR)^{\times}.
$$
Both sides have order $D_n=(q-1)q^{n-1}$, so it is an isomorphism. In particular every finite layer is abelian and totally ramified.
For an example, take $K=\mathbb Q_p$, $\pi=p$, and the equally admissible series $f(T)=(1+T)^p-1$. Its <formal group law> is the <formal multiplicative group> $F(X,Y)=X+Y+XY$, with $[a](T)=(1+T)^a-1$ for $a\in\mathbb Z_p$. The <binomial polynomials> make its coefficients integral. Its level-$n$ torsion points are $\zeta-1$ with $\zeta^{p^n}=1$. Thus the <Lubin–Tate cyclotomic example> gives $K_{p,n}=\mathbb Q_p(\mu_{p^n})$ and Galois action $\zeta\mapsto\zeta^a$. The order formula includes $p=2,n=1$, when the first field and its <Galois group> are trivial over $\mathbb Q_2$.
Finally choose primitive points compatibly, $[\pi]\lambda_{n+1}=\lambda_n$. They exist by taking a root of $f(T)-\lambda_n$ in the algebraic closure; its roots have positive <valuation> and exact level $n+1$. Then $K_{\pi,n}\subseteq K_{\pi,n+1}$, and restriction of Galois automorphisms corresponds to reduction of <unit> classes. The <Lubin–Tate tower> $K_\pi=\bigcup_nK_{\pi,n}$ consequently has the topological <Galois group>
$$
\boxed{\operatorname{Gal}(K_\pi/K)\cong\varprojlim_n(R/\pi^nR)^{\times}\cong R^{\times}.}
$$
The final isomorphism uses completeness of the <discrete valuation ring> $R$. This computes the finite-layer and infinite-tower Galois groups directly from the formal action and the primitive Eisenstein <polynomials>.
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