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\section{Formal groups and Lubin--Tate
modules}\label{formal-groups-and-lubintate-modules}

\emph{Written by OpenAI GPT-6.1 Sol in Codex, Ultra effort, October
2026. Self-checked by the writing AI; no independent review is claimed.
Public domain (CC0).}

The multiplicative expression \(X+Y+XY\) becomes a group law near zero
because \(1+(X+Y+XY)=(1+X)(1+Y)\). A formal group retains such a law as
a power series, before choosing the elements on which it will be
evaluated. Lubin--Tate theory builds one whose multiplication by a
uniformizer reduces to Frobenius. Its division points will furnish
explicit abelian extensions in the next lesson.

Two congruences drive the construction. The linear coefficient is a
uniformizer \(\pi\), while reduction of the entire series is \(X^q\).
One controls the coefficient to be solved; the other makes the
obstruction divisible by \(\pi\). We prove that coefficient calculation,
derive the group and its scalar action, and then compare different
uniformizers using Frobenius on the completed unramified tower.

We use
\href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/proof-dependencies.html\#NT-LOC-02}{Completions,
the p-adic numbers and complete discretely valued fields}, Theorem 2.1
and Proposition 2.2, for completion, extension of isometries and integer
approximation in \(\mathbf Z_p\);
\href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/proof-dependencies.html\#NT-LOC-04}{Extensions
of complete valued fields}, Theorem 1.2, for completeness of finite
extensions and their uniquely extended valuation; and
\href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/proof-dependencies.html\#NT-LOC-07}{Unramified
and totally ramified extensions}, Theorem 2.1 and Corollary 3.1, for the
unramified tower and its arithmetic Frobenius. The construction is
algebraic in its power-series variables; convergence will enter only
when we evaluate them or complete a coefficient ring.

\textbf{Prerequisite proof availability.} The named results below
identify specific programme lessons. The
\href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/proof-dependencies.html\#lesson-7}{prerequisite
record} distinguishes published proofs, supplied owner texts awaiting
publication, and missing full proofs. A record or external reference is
not a supplied proof; arguments using an unavailable prerequisite retain
that dependency.

\subsection{1. Formal laws and their
inverses}\label{formal-laws-and-their-inverses}

Let \(R\) be a commutative ring with 1. A \textbf{one-dimensional
commutative formal group law} is \(F(X,Y)\in R[[X,Y]]\) with

\[
\begin{gathered}
F(X,Y)=X+Y+\text{terms of total degree at least 2},\\
F(X,0)=X,\quad F(0,Y)=Y,\quad F(X,Y)=F(Y,X),\\
F(F(X,Y),Z)=F(X,F(Y,Z)).
\end{gathered}
\tag{1}
\]

Substitution of series with zero constant term is well defined: a
coefficient of a given total degree involves only finitely many terms.
No analytic convergence is needed for these identities.

A series \(h(X)=cX+\cdots\) with \(c\in R^\times\) has a unique
compositional inverse. Solve \(h(k(X))=X\) degree by degree:
\(k_1=c^{-1}\), and at degree \(m\) the unknown coefficient of \(k\) has
coefficient \(c\), hence has a unique value. The same recursion
constructs a left inverse; associativity of substitution identifies left
and right inverses. If one starts with the linear, commutative and
associative conditions alone, the identity conditions in (1) follow:
\(h(X)=F(X,0)\) satisfies \(h\circ h=h\), and its invertible linear
coefficient is 1, so composing with its inverse gives \(h=X\). The other
identity follows by commutativity.

\subsubsection{Proposition 7.1. The formal group
inverse}\label{proposition-7.1.-the-formal-group-inverse}

There is a unique \(\iota_F(X)=-X+\cdots\in R[[X]]\) satisfying

\[
F(X,\iota_F(X))=0.
\tag{2}
\]

It also satisfies \(F(\iota_F(X),X)=0\).

\textbf{Proof.} The degree-one term forces \(-X\). Once coefficients
below degree \(m\) are chosen, adding a new term \(bX^m\) to that
truncation adds exactly \(bX^m\) to \(F(X,\iota_F(X))\) in degree \(m\).
The coefficient comes from the linear \(Y\) term; every nonlinear term
involving the change has higher degree. Choose \(b\) to cancel the known
coefficient. This recursion works over any \(R\), proves uniqueness, and
commutativity proves the other identity. \(\square\)

If \(S\) is a separated complete \(R\)-algebra for the powers of an
ideal \(I\), evaluation on \(x,y\in I\) converges in the \(I\)-adic
topology. Its degree-\(m\) terms lie in \(I^m\). The identities (1)--(2)
make \(I\) an abelian group, denoted \(F(I)\), with operation
\(x+_F y=F(x,y)\). In a finite extension of a complete valued field,
integral coefficients likewise converge on its maximal ideal, since
terms of increasing degree have valuation tending to infinity.

The additive law is \(G_a(X,Y)=X+Y\). The multiplicative law is
\(G_m(X,Y)=X+Y+XY\); associativity and commutativity follow from
multiplication of \(1+X,1+Y,1+Z\), and its inverse is \(-X/(1+X)\).

A homomorphism \(h:F\to H\) is a zero-constant series satisfying \[
h(F(X,Y))=H(h(X),h(Y)).
\] It is an isomorphism if its linear coefficient is a unit, since its
compositional inverse then satisfies the reverse homomorphism identity
by substitution. The endomorphisms of \(F\) form a ring, with sum
\(F(h(X),k(X))\), product \(h\circ k\), zero series 0 and identity
\(X\). Associativity of \(F\) supplies additive associativity;
\(\iota_F\circ h\) supplies the negative; the homomorphism identity and
associativity of composition give distributivity. This ring need not be
commutative for an arbitrary formal group.

\subsection{2. The coefficient lemma}\label{the-coefficient-lemma}

Let \(K\) be a nonarchimedean local field, \(\mathcal O=\mathcal O_K\),
\(\pi\) a uniformizer, and \(\mathcal O/\pi\mathcal O=\mathbf F_q\). A
\textbf{Lubin--Tate series for \(\pi\)} is \(f\in\mathcal O[[X]]\) with

\[
f(X)\equiv\pi X\pmod{\text{degree }2},
\qquad f(X)\equiv X^q\pmod\pi.
\tag{3}
\]

Write \(\mathcal F_\pi\) for their set. The series \(\pi X+X^q\) is an
example.

\subsubsection{Lemma 7.2. The Lubin--Tate coefficient
lemma}\label{lemma-7.2.-the-lubintate-coefficient-lemma}

For \(f,g\in\mathcal F_\pi\) and \(L(X_1,\ldots,X_d)=\sum_i a_iX_i\),
\(a_i\in\mathcal O\), there is a unique zero-constant series
\(H\in\mathcal O[[X_1,\ldots,X_d]]\) with linear part \(L\) and

\[
f(H(X))=H(g(X_1),\ldots,g(X_d)).
\tag{4}
\]

\textbf{Proof.} Suppose a polynomial \(H_{<m}\) with the prescribed
linear part satisfies (4) through degrees below \(m\), where \(m\geq2\).
Let \(E_m\) be the homogeneous degree-\(m\) part of the left side minus
the right side. Its coefficients are divisible by \(\pi\). Indeed,
reducing either side modulo \(\pi\) gives \(\overline H_{<m}(X)^q\) and
\(\overline H_{<m}(X_1^q,\ldots,X_d^q)\), which agree: in characteristic
\(p\) the \(q\)-th power respects sums, and each coefficient in
\(\mathbf F_q\) satisfies \(a^q=a\).

Adding a homogeneous polynomial \(C_m\) of degree \(m\) changes this
error in degree \(m\) by

\[
(\pi-\pi^m)C_m.
\]

On the left only the linear coefficient of \(f\) contributes the change
in that degree; on the right substitute the linear terms \(\pi X_i\)
into \(C_m\). Nonlinear terms contribute only to later degrees. The
unique correction is

\[
C_m=-\frac{E_m}{\pi(1-\pi^{m-1})}.
\tag{5}
\]

It has integral coefficients because \(E_m\) is divisible by \(\pi\) and
\(1-\pi^{m-1}\) is a unit. The initial linear equation holds because
multiplication by \(\pi\) commutes with \(L\). Induction constructs
every homogeneous part and proves (4). If two solutions first differ in
degree \(m\), their difference there is killed by \(\pi-\pi^m\), a
nonzero element of the domain \(\mathcal O\); hence that difference is
zero. This also proves uniqueness. \(\square\)

The coefficients are determined by algebraic degree induction. We are
not substituting large elements into the series or assuming that \(\pi\)
is an invertible coefficient.

\textbf{Computing the first nonlinear part.} For \(f=g=\pi X+X^q\), let
\(F\) be the solution with linear part \(X+Y\). All homogeneous parts of
degrees \(2,\ldots,q-1\) are zero: the initial error has no terms in
those degrees, and (5) then gives zero at each step. In degree \(q\) the
initial error is \((X+Y)^q-X^q-Y^q\). Therefore \[
F(X,Y)=X+Y+
\frac{\displaystyle\sum_{j=1}^{q-1}\binom qj X^jY^{q-j}}
     {\pi^q-\pi}
\pmod{\text{degree }q+1}.
\] Each displayed coefficient is integral: reduction modulo \(\pi\)
kills the numerator, and \((\pi^q-\pi)/\pi\) is a unit. In equal
characteristic the numerator is identically zero. Indeed, in that case
the full additive law commutes with \(\pi X+X^q\), so uniqueness proves
\(F=X+Y\) in every degree. Over \(\mathbf Q_2\) with \(\pi=2\), the
formula gives the coefficient 1 of \(XY\). Here \(2X+X^2=(1+X)^2-1\)
commutes with \(X+Y+XY\), and uniqueness proves that this is the full
law. The same coefficient construction thus yields different group laws
in the two characteristics.

\subsection{3. The group and its scalar
action}\label{the-group-and-its-scalar-action}

For \(f\in\mathcal F_\pi\), apply the lemma with two variables, source
series \(f\) and linear form \(X+Y\). Denote the resulting series by
\(F_f(X,Y)\). For \(f,g\in\mathcal F_\pi\) and \(a\in\mathcal O\), let
\([a]_{f,g}\) be the one-variable solution with target \(f\), source
\(g\), and linear term \(aX\). Put \([a]_f=[a]_{f,f}\).

\subsubsection{Theorem 7.3. Lubin--Tate formal
modules}\label{theorem-7.3.-lubintate-formal-modules}

The series \(F_f\) is a formal group law, and

\[
\mathcal O\longrightarrow\operatorname{End}_{\mathcal O}(F_f),
\qquad a\longmapsto[a]_f
\tag{6}
\]

is an injective ring homomorphism with \([\pi]_f=f\). Each
\([a]_{f,g}:F_g\to F_f\) is a homomorphism compatible with these scalar
actions. When \(a\) is a unit it is an isomorphism. In particular, all
\(F_f\), \(f\in\mathcal F_\pi\), are isomorphic over \(\mathcal O\).

\textbf{Proof.} In the following comparisons, the coefficient lemma
applies to the indicated linear part and the indicated source series.

The two series \(F_f(X,Y)\) and \(F_f(Y,X)\) satisfy (4) with target
\(f\), source \(f\) and linear part \(X+Y\). They are equal. The
specialization \(F_f(X,0)\) and the identity \(X\) both commute with
\(f\) and have linear term \(X\), so they agree. Finally,
\(F_f(F_f(X,Y),Z)\) and \(F_f(X,F_f(Y,Z))\) both commute with
simultaneous substitution of \(f\) in the three variables and have
linear part \(X+Y+Z\). Uniqueness gives associativity. Thus \(F_f\)
satisfies (1), and Proposition 7.1 supplies its inverse.

The homomorphism identity for \(h=[a]_{f,g}\) follows by comparing \[
h(F_g(X,Y))\quad\text{and}\quad F_f(h(X),h(Y)).
\] Both have linear part \(a(X+Y)\), target \(f\) and simultaneous
source \(g\). For the first, commute \(h\) with \(g,f\) and use the
defining identity for \(F_g\); for the second, use it for \(F_f\). The
lemma makes them equal.

Likewise, the identities

\[
\begin{aligned}
[a+b]_{f,g}&=F_f([a]_{f,g},[b]_{f,g}),\\
[a]_{f,g}\circ[b]_{g,h}&=[ab]_{f,h}
\end{aligned}
\tag{7}
\]

follow by comparing their linear parts and commuting equations.
Explicitly, for the composition,
\(f[a]_{f,g}[b]_{g,h}=[a]_{f,g}g[b]_{g,h}=[a]_{f,g}[b]_{g,h}h\); the sum
commutes because \(f\) is an endomorphism of \(F_f\). The solutions of
linear parts \(0,X,\pi X\) for target and source \(f\) are respectively
\(0,X,f\). These facts prove the ring assertions in (6); injectivity
follows by reading the linear coefficient.

Equation (7) also gives \([a]_{f,g}[b]_g=[ab]_{f,g}=[b]_f[a]_{f,g}\), so
scalar actions are respected. If \(a\) is a unit, \([a^{-1}]_{g,f}\) is
the inverse by (7). Taking \(a=1\) gives the final isomorphism
assertion. \(\square\)

A formal \(\mathcal O\)-module means a formal group with an
\(\mathcal O\)-action whose scalar \(a\) has linear coefficient \(a\).
The construction is unique for prescribed \(f=[\pi]\): its law must
satisfy (4) with linear part \(X+Y\), and its scalar series must commute
with \([\pi]\) and have the prescribed linear coefficient. The lemma
determines both. We write \(F_f[\pi^n]\) for its \(\pi^n\)-division
points when the series are evaluated in algebraic maximal ideals.

\subsection{4. Frobenius on the completed unramified
ring}\label{frobenius-on-the-completed-unramified-ring}

Let \(\widehat{K^{\mathrm{ur}}}\) be the completion of the maximal
unramified extension and \(\widehat{\mathcal O}\) its valuation ring.
The uniformizer remains \(\pi\), its residue field is
\(\overline{\mathbf F}_q\), and its value group is \(\mathbf Z\). To see
that completion does not alter these assertions, a Cauchy sequence
representing a nonzero element of the completion has eventually constant
valuation, and an integral Cauchy sequence has eventually constant
residue. Conversely every algebraic residue element appears in a finite
unramified extension. These observations also show that
\(\widehat{\mathcal O}\) is a complete discrete valuation ring.

Arithmetic Frobenius is an isometry on the unramified union, so it
extends, together with its inverse, to an automorphism \(\varphi\) of
the completion. It fixes \(K\) and \(\pi\), and reduces to
\(x\mapsto x^q\).

\subsubsection{Lemma 7.4. Additive and multiplicative Frobenius
equations}\label{lemma-7.4.-additive-and-multiplicative-frobenius-equations}

The maps

\[
x\longmapsto\varphi(x)-x
\quad\text{on }\widehat{\mathcal O},\qquad
x\longmapsto\varphi(x)/x
\quad\text{on }\widehat{\mathcal O}^{\times}
\tag{8}
\]

are surjective. Their kernels are \(\mathcal O_K\) and
\(\mathcal O_K^\times\), respectively.

\textbf{Proof.} For an additive target \(c\), solve \(y^q-y=\bar c\) in
the algebraically closed residue field. Lift a solution to \(x_0\). If
\(\varphi(x_r)-x_r-c\in\pi^r\widehat{\mathcal O}\), solve \(y^q-y=b\),
where \(b\) is the negative residue of \(\pi^{-r}(\varphi(x_r)-x_r-c)\).
A correction \(\pi^r y\) removes that error modulo \(\pi^{r+1}\).
Inductively the error tends to zero, the corrections tend to zero, and
completeness gives an exact solution.

For a unit target \(c\), first solve \(y^{q-1}=\bar c\) with \(y\ne0\)
and lift to a unit \(x_0\). Its quotient \(\varphi(x_0)/x_0\) matches
\(c\) modulo \(\pi\). If the residual target ratio is \(1+\pi^r b\)
modulo \(\pi^{r+1}\), multiply the tentative solution by \(1+\pi^r y\).
Its Frobenius ratio is \[
1+\pi^r(\varphi(y)-y)\pmod{\pi^{r+1}}.
\] Solving \(y^q-y=\bar b\) improves the agreement. The successive unit
corrections form a convergent product, giving an exact preimage of
\(c\).

For the kernels, suppose \(\varphi(x)=x\), \(x\in\widehat{\mathcal O}\).
Its residue lies in \(\mathbf F_q\), so choose \(a_0\in\mathcal O_K\)
with that residue. Then \(x=a_0+\pi x_1\), and \(x_1\) is again fixed.
Repeat to express \(x\) as the convergent series
\(\sum_{r\geq0}\pi^r a_r\), with all \(a_r\) chosen in a fixed finite
set of residue representatives in \(\mathcal O_K\). Completeness of
\(K\) puts its limit in \(\mathcal O_K\). The reverse inclusion is fixed
by \(\varphi\). For a unit, the same assertion gives precisely
\(\mathcal O_K^\times\). \(\square\)

It is the completion that guarantees convergence of the corrections.
Their residues can lie in larger and larger finite fields, so there is
no reason for the limit to belong to the algebraic unramified union.

\subsection{5. Changing the uniformizer}\label{changing-the-uniformizer}

We need a twisted version of the coefficient lemma. Let \(\pi,\pi'\) be
uniformizers, \(f\in\mathcal F_\pi\), \(f'\in\mathcal F_{\pi'}\), and
let \(L(X)\) be a linear form over \(\widehat{\mathcal O}\) such that \[
\pi'L=\pi L^\varphi.
\tag{9}
\] Superscript \(\varphi\) means that it acts on coefficients, leaving
variables alone. Then there is a unique series \(H\) with linear part
\(L\) satisfying \[
f'(H(X))=H^\varphi(f(X_1),\ldots,f(X_d)).
\tag{10}
\]

Here is the full additional coefficient calculation. In degree
\(m\geq2\), the error is divisible by \(\pi'\): modulo the maximal ideal
the two sides are \(\overline H(X)^q\) and
\(\overline H^\varphi(X_1^q,\ldots,X_d^q)\), equal because \(\varphi\)
raises residue coefficients to the \(q\)-th power. For a correction
coefficient \(a\), the equation is

\[
\pi'a-\pi^m\varphi(a)=-e_m,\qquad e_m\in\pi'\widehat{\mathcal O}.
\]

Put \(\alpha=\pi^m/\pi'\), \(b=-e_m/\pi'\). Since \(v(\alpha)=m-1>0\),
the unique solution of \(a-\alpha\varphi(a)=b\) is

\[
a=\sum_{j\geq0}\alpha^j\varphi^j(b).
\tag{11}
\]

The coefficients \(\alpha\) are fixed by \(\varphi\), so substitution
telescopes and proves the equation. The terms tend to zero and the ring
is complete. A difference of two solutions would satisfy
\(a=\alpha\varphi(a)\), impossible for a nonzero element because
\(\varphi\) preserves valuation. This proves existence and uniqueness in
every homogeneous degree. Condition (9) is exactly the degree-one
equation. We have proved (10) without appealing to a formal logarithm.

\subsubsection{Theorem 7.5. Comparison over the completed unramified
extension}\label{theorem-7.5.-comparison-over-the-completed-unramified-extension}

Suppose \(\pi'=u\pi\), \(u\in\mathcal O_K^\times\). For any
\(f\in\mathcal F_\pi\) and \(f'\in\mathcal F_{\pi'}\), there is an
isomorphism of formal \(\mathcal O_K\)-modules

\[
\theta:F_f\longrightarrow F_{f'}
\quad\text{over }\widehat{\mathcal O}
\]

with linear term \(\varepsilon X\), where \(\varepsilon\) is a unit
satisfying \[
\varphi(\varepsilon)=u\varepsilon.
\] It satisfies

\[
\theta^\varphi=\theta\circ[u]_f.
\tag{12}
\]

\textbf{Proof.} Lemma 7.4 supplies \(\varepsilon\). Equation (9) holds
for \(L=\varepsilon X\), so the twisted lemma constructs \(\theta\)
satisfying \(f'\theta=\theta^\varphi f\).

Compare \(\theta(F_f(X,Y))\) and \(F_{f'}(\theta(X),\theta(Y))\). The
coefficients of \(F_f,F_{f'}\) are fixed by \(\varphi\). Their commuting
identities and \(f'\theta=\theta^\varphi f\) therefore show that both
compared series satisfy (10), with linear part \(\varepsilon(X+Y)\).
Uniqueness proves the formal group homomorphism identity. Likewise,
\(\theta[a]_f\) and \([a]_{f'}\theta\) satisfy (10) with linear term
\(\varepsilon aX\), so they agree for every \(a\in\mathcal O_K\). Since
\(\varepsilon\) is a unit, the formal compositional inverse exists and
makes \(\theta\) an isomorphism.

Finally, \(\theta^\varphi\) satisfies (10) by applying \(\varphi\) to
its defining equality. The series \(\theta[u]_f\) satisfies the same
equation because \([u]_f\) commutes with \(f\) and has fixed
coefficients. Their linear terms agree,
\(\varphi(\varepsilon)X=\varepsilon uX\), so uniqueness gives (12).
\(\square\)

The direction matters: with \(\pi'=u\pi\), the source of \(\theta\) is
the group for \(\pi\), and \([u]_f\) in (12) acts on that source.
Inverting the direction changes the corresponding unit equation.

\subsection{\texorpdfstring{6. The multiplicative example over
\(\mathbf Z_p\)}{6. The multiplicative example over \textbackslash mathbf Z\_p}}\label{the-multiplicative-example-over-mathbf-z_p}

For \(a\in\mathbf Z_p\), the binomial series

\[
[a](X)=(1+X)^a-1=\sum_{j\geq1}\binom aj X^j
\tag{13}
\]

has integral coefficients. For fixed \(j\), \(\binom aj\) is a
continuous polynomial function of \(a\) over \(\mathbf Q_p\).
Approximate \(a\) by nonnegative integers, whose binomial coefficients
are integral; closedness of \(\mathbf Z_p\) proves the assertion.

For nonnegative integers, (13) respects \(G_m\) by the ordinary power
identity. Taking coefficientwise limits proves the homomorphism identity
for \(a\in\mathbf Z_p\), since each coefficient of a substituted
identity depends on finitely many coefficients. The same reasoning
applied to integer pairs gives \[
[a+b]=G_m([a],[b]),\qquad [ab]=[a]\circ[b].
\] The series \(f=(1+X)^p-1\) has linear term \(pX\) and reduction
\(X^p\), so belongs to \(\mathcal F_p\). The group \(G_m\) and its
series (13) satisfy the defining equations of Theorem 7.3; uniqueness
identifies them with \(F_f\) and \([a]_f\). The other choice \(pX+X^p\)
therefore yields a formal group isomorphic to \(G_m\) over
\(\mathbf Z_p\).

Every endomorphism \(h\) of \(G_m\) over \(\mathbf Z_p\) commutes with
its integer multiplication \([p]\), because a homomorphism preserves a
repeated group sum. If its linear coefficient is \(a\), Lemma 7.2 makes
it the unique series commuting with \(f=[p]\) and starting with \(aX\).
Thus it is (13), proving

\[
\operatorname{End}_{\mathbf Z_p}(G_m)=\mathbf Z_p
\]

with the ring operations already described. This argument does not
assert an analogous derivative classification for arbitrary formal
groups in positive characteristic.

\subsection{Exercises}\label{exercises}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \textbf{Easy.} Verify the formal group identities for \(G_m\), its
  inverse, and the integral scalar series \((1+X)^a-1\),
  \(a\in\mathbf Z_p\).
\item
  \textbf{Medium.} Construct the formal inverse of an arbitrary \(F\)
  over an arbitrary commutative ring, proving uniqueness without
  division by integers.
\item
  \textbf{Medium.} Prove that every endomorphism of \(G_m\) over
  \(\mathbf Z_p\) is \((1+X)^a-1\) for a unique \(a\in\mathbf Z_p\).
\item
  \textbf{Hard.} Prove both surjectivities in Lemma 7.4, identify both
  kernels, and explain why completing the unramified union is part of
  the argument.
\end{enumerate}

\subsection{Solutions}\label{solutions}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Multiplying \(1+X,1+Y\) yields \(1+G_m(X,Y)\), so commutativity,
  associativity and the identity follow from multiplication in
  \(R[[X,Y,Z]]\). The inverse is \(-X/(1+X)=-X+X^2-\cdots\). For fixed
  \(j\), the continuous polynomial \(\binom aj\) takes integral values
  at nonnegative integers dense in \(\mathbf Z_p\); its value at \(a\)
  is therefore integral. The integer power identity gives the scalar
  homomorphism identity, and coefficientwise continuity extends it to
  every \(a\). The same extension for addition and composition gives the
  scalar ring action.
\item
  Write the desired inverse as \(-X+\sum_{m\geq2}b_mX^m\). After
  coefficients below \(m\) are set, the degree-\(m\) coefficient of
  \(F(X,\iota(X))\) is a known element plus \(b_m\). A new \(b_mX^m\)
  entering a nonlinear term has degree at least \(m+1\), so its only
  contribution at degree \(m\) comes from the linear \(Y\). Set \(b_m\)
  to the negative of the known coefficient. This recursion uses no
  division and gives existence and uniqueness in every degree.
  Commutativity gives the left inverse identity as well.
\item
  An endomorphism preserves \(p\) repeated sums, hence commutes with
  \(f=(1+X)^p-1\). Its linear coefficient \(a\) lies in \(\mathbf Z_p\).
  Lemma 7.2 with one variable says that exactly one series with linear
  part \(aX\) commutes with \(f\). The integral series (13) is such a
  series, so it is the endomorphism. Conversely all those series are
  endomorphisms by solution 1. Linear coefficients distinguish \(a\),
  and the addition and composition identities identify the endomorphism
  ring with \(\mathbf Z_p\).
\item
  Reduce an additive target modulo \(\pi\) and solve \(y^q-y=c\) in
  \(\overline{\mathbf F}_q\); successive corrections \(\pi^r y_r\) solve
  the next residue error, producing a Cauchy sequence. For a unit
  target, first solve \(y^{q-1}=c\) with \(y\ne0\), then correct with
  factors \(1+\pi^r y_r\). Their Frobenius quotients have residue
  correction \(y_r^q-y_r\), so this also improves the error one depth at
  a time. The product is Cauchy and remains a unit; continuity gives the
  exact solution. A fixed integral element has residue in
  \(\mathbf F_q\); subtract its ground-field representative, divide by
  \(\pi\), and repeat. Its resulting ground-field digit series converges
  in \(K\), proving the additive kernel is \(\mathcal O_K\), and the
  unit kernel is \(\mathcal O_K^\times\). The correction residues need
  not stay in one finite residue extension, so their limit is guaranteed
  in the completion, not in the uncompleted algebraic union.
\end{enumerate}

\subsection{Editable edition}\label{editable-edition}

The reading edition provides the complete LaTeX source of this lesson,
the cumulative course LaTeX and the editable source ZIP. The archive
contains all twenty-four Markdown lessons, complete LaTeX bodies,
original diagrams, metadata and reproduction instructions.

\subsection{References}\label{references}

Sections 1--3 prove inversion, coefficient correction, every
formal-group identity and scalar compatibility. The first nonlinear
coefficients are computed explicitly. Sections 4--6 prove the Frobenius
equations and the twisted construction over the completed unramified
ring.

\begin{itemize}
\tightlist
\item
  \href{https://www.jmilne.org/math/CourseNotes/CFT.pdf}{J. S. Milne,
  Class Field Theory, version 4.03}.
\item
  \href{https://arxiv.org/abs/math/0606108v2}{Teruyoshi Yoshida, Local
  class field theory via Lubin--Tate theory, arXiv:math/0606108v2}.
\end{itemize}

The \href{../free-proofs.html}{proof guide} gives the lesson sequence
and the exact prerequisite record. External references accompany the
written arguments.

\end{document}