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\section{Kronecker--Weber and the maximal abelian extension of the
rationals}\label{kroneckerweber-and-the-maximal-abelian-extension-of-the-rationals}

\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).}

\textbf{Lesson 20.} Every finite abelian extension of the rationals is
contained in a cyclotomic field. The ray fields of \(\mathbf Q\) make
this theorem a direct consequence of global class field theory. We
identify those fields, compute the Artin map on arbitrary rational
idèles, and derive the conductor--discriminant formula. These
identifications also specify the Galois action on the roots of unity
appearing in the Bost--Connes system.

We use
\href{../ray-class-fields-conductors-and-ideal-reciprocity.html}{Ray
class fields, conductors and ideal reciprocity}, the explicit local
cyclotomic symbol in Corollary 9.5 of
\href{../explicit-local-reciprocity-and-existence.html}{Explicit local
reciprocity and existence}, and the local conductor--discriminant
theorem, Theorem 10.5 of
\href{../abelian-ramification-conductors-and-hasse-arf.html}{Abelian
ramification, conductors and Hasse--Arf}. Theorem 12.1 and Theorem 12.3
of
\href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/proof-dependencies.html\#NT-ANT-12}{Cyclotomic
fields} prove cyclotomic irreducibility, degrees, inertia and arithmetic
Frobenius.
\href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/proof-dependencies.html\#NT-ADL-06}{Quasi-characters
and Hecke characters}, especially Theorem 6.3, supplies the character
terminology and its ideal/idèle convention. All these providers are
written lessons.

Fix \(\zeta_m=e^{2\pi i/m}\), put \(U_m=(\mathbf Z/m\mathbf Z)^\times\),
and write \[
\sigma_a(\zeta_m)=\zeta_m^a\quad(a\in U_m).
\tag{1}
\] The groups \(U_1,U_2\) are trivial. Arithmetic reciprocity sends an
unramified uniformizer to arithmetic Frobenius. Geometric reciprocity is
its inverse. The cyclotomic character always records the exponent in
(1); its definition does not change when reciprocity is inverted.

\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-20}{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. The rational ray class
fields}\label{the-rational-ray-class-fields}

For an integer \(m\geq1\), the moduli \(m\infty\) and \(m\) differ by
the real place condition. In the first, principal generators must be
positive; in the second, either sign is allowed.

\textbf{Proposition 20.1.} Their ray groups and fields are \[
\begin{aligned}
\operatorname{Cl}_{m\infty}(\mathbf Q)&\simeq U_m,&
\mathbf Q_{m\infty}&=\mathbf Q(\zeta_m),\\
\operatorname{Cl}_{m}(\mathbf Q)&\simeq U_m/\langle-1\rangle,&
\mathbf Q_m&=\mathbf Q(\zeta_m)^+.
\end{aligned}
\tag{2}
\] Here the superscript \(+\) means the subfield fixed by complex
conjugation. The formulas include \(m=1,2\).

\textbf{Proof.} Every fractional ideal of \(\mathbf Q\) prime to \(m\)
has a unique positive rational generator \(a\), whose numerator and
denominator are coprime to \(m\). Send that ideal to \(a\bmod m\). Every
unit residue occurs, by taking a positive integral representative. Its
kernel consists exactly of the ideals with positive generator congruent
to \(1\) modulo \(m\), giving the first group in (2). If the generator
may have either sign, an ideal is ray principal exactly when its
positive generator is congruent to \(1\) or \(-1\); this gives the
second group.

Let \(E=\mathbf Q(\zeta_m)\). The local symbol proves its conductor
divides \(m\infty\). Indeed, write \(m=p^a b\), \(p\nmid b\). At \(p\),
the field generated by the \(b\)-th roots is unramified, so local units
act trivially on them. Units congruent to \(1\pmod{p^a}\) act trivially
on the \(p^a\)-th roots by Corollary 9.5. When \(a=0\), all units
suffice. At the real place, positive numbers act trivially. Thus the
entire modulus subgroup is in the global reciprocity kernel, by local
compatibility. Theorem 18.4 places \(E\) in \(\mathbf Q_{m\infty}\).
Both degrees are \(\varphi(m)\), the first by cyclotomic irreducibility
and the second by the ray-group computation. They are equal.

Dropping the real place adds the class of a negative real component to
the norm subgroup. Its symbol on \(E\) is complex conjugation: local
reciprocity at \(\mathbf R\) identifies the sign quotient with
\(\operatorname{Gal}(\mathbf C/\mathbf R)\). Therefore the corresponding
subfield is \(E^+\). Its group is the quotient by \(-1\) in (1). This
proves the remaining field assertion, also when conjugation is already
trivial. \(\square\)

The modulus in (2) need not be the field's least conductor. For example
\(\mathbf Q(\zeta_2)=\mathbf Q\), and for odd \(m\) the fields of orders
\(m\) and \(2m\) are equal. A conductor is a minimal condition, whereas
a ray modulus is a specified condition.

\subsection{2. The Kronecker--Weber
theorem}\label{the-kroneckerweber-theorem}

\textbf{Theorem 20.2 (Kronecker--Weber).} Every finite abelian
\(L/\mathbf Q\) lies in \(\mathbf Q(\zeta_m)\) for some positive integer
\(m\). Consequently \[
\mathbf Q^{\mathrm{ab}}=\mathbf Q^{\mathrm{cycl}}
=\bigcup_{m\geq1}\mathbf Q(\zeta_m)=\mathbf Q(\mu_\infty).
\tag{3}
\]

\textbf{Proof.} The extension has a finite global conductor, by
Proposition 18.3. Enlarge it, if necessary, to a modulus \(m\infty\).
The ray-field containment theorem and (2) give
\(L\subseteq\mathbf Q_{m\infty}=\mathbf Q(\zeta_m)\). Every cyclotomic
field is abelian by (1), so their union is contained in the maximal
abelian extension. The reverse inclusion follows from the finite
assertion, since every algebraic element in \(\mathbf Q^{\mathrm{ab}}\)
belongs to a finite abelian extension. \(\square\)

This proof identifies the containing field through its ray conditions.
Section 7 gives a second proof using local Kronecker--Weber and finite
inertia groups.

\subsection{3. The explicit rational Artin
map}\label{the-explicit-rational-artin-map}

The normalization of idèles in lesson 17 gives a topological isomorphism
\[
C_{\mathbf Q}\simeq\mathbf R_{>0}\times\widehat{\mathbf Z}^{\times}.
\tag{4}
\] Explicitly, for an idèle \(x=(x_\infty,(x_p))\), set \[
q=\prod_p p^{-v_p(x_p)}>0,\quad
\epsilon=\operatorname{sgn}(x_\infty),\quad
t=\epsilon qx_\infty>0,\quad u_p=\epsilon qx_p\in\mathbf Z_p^\times.
\tag{5}
\] Only finitely many factors define \(q\). Multiplication by the
principal idèle \(\epsilon q\) yields the normalized representative
\((t,u)\). A rational number that is a unit at every prime is \(\pm1\),
and positivity fixes its sign; hence the representative is unique.
Continuity holds on the restricted-product neighborhoods where the
finitely many valuations and the real sign are fixed.

\textbf{Proposition 20.3.} For \(u\in\widehat{\mathbf Z}^{\times}\),
compatible exponent notation gives \[
\operatorname{rec}_{\mathrm{arith}}(t,u)=\sigma_{u^{-1}},
\qquad
\operatorname{rec}_{\mathrm{geom}}(t,u)=\sigma_u.
\tag{6}
\] For a prime \(p\), the idèle \(\varpi_p\) equal to \(p\) at \(p\) and
\(1\) elsewhere acts on \(\mathbf Q(\zeta_m)\), \(p\nmid m\), as
\(\sigma_p\) in the arithmetic convention. The idèle equal to \(-1\) at
the real place and \(1\) at every finite place acts as complex
conjugation.

\textbf{Proof.} A positive real component acts trivially by real local
reciprocity. On \(p^a\)-th roots, the local component \(u_p\) acts with
exponent \(u_p^{-1}\), by Corollary 9.5; every other finite unit
component is unramified on those roots and acts trivially. Combining the
prime-power roots gives the first formula on \(\mathbf Q(\zeta_m)\), for
every \(m\). Local compatibility and (3) make it the full global
formula. Inverting gives geometric reciprocity.

For \(\varpi_p\), (5) gives \(t=1/p\), \(u_p=1\), and \(u_\ell=1/p\) for
\(\ell\ne p\). Thus \(u^{-1}\equiv p\pmod m\) when \(p\nmid m\), proving
the Frobenius assertion. No such residue \(p\pmod m\) defines a unit
when \(p\mid m\); at the \(p\)-primary roots this particular idèle acts
trivially, and at the prime-to-\(p\) roots it acts with exponent \(p\).

For the negative real idèle, (5) gives \(t=1\), \(u_p=-1\) at every
finite prime. Its inverse is itself, so (6) is \(\sigma_{-1}\), complex
conjugation. \(\square\)

\textbf{Corollary 20.4.} The cyclotomic character is a canonical
topological isomorphism \[
\chi_{\mathrm{cyc}}:\operatorname{Gal}(\mathbf Q^{\mathrm{ab}}/\mathbf Q)
\xrightarrow{\sim}\widehat{\mathbf Z}^{\times},
\qquad \sigma_a\longmapsto a.
\tag{7}
\] Arithmetic reciprocity induces an isomorphism from
\(C_{\mathbf Q}/\mathbf R_{>0}\), and its composite with (7) is
\(u\mapsto u^{-1}\).

\textbf{Proof.} The finite isomorphisms (1) are compatible with
reduction of exponents. Taking inverse limits, as proved in Proposition
1.3, gives (7) for the cyclotomic union, which is (3). Formula (6) is
onto and has kernel exactly the positive real factor: a unit tuple
acting trivially on every root has residue \(1\) modulo every integer.
Both (7) and its inverse are continuous by their finite quotient
coordinates. \(\square\)

\subsection{4. Dirichlet, Galois and Hecke
characters}\label{dirichlet-galois-and-hecke-characters}

A Dirichlet character means a character of some \(U_m\), with characters
identified when they are inflations of the same primitive character. Its
\textbf{finite conductor} \(f(\chi)\) is the least modulus of that
primitive character. When values on integers are used, the primitive
character is extended by zero precisely at integers not coprime to
\(f(\chi)\).

\textbf{Proposition 20.5.} Dirichlet characters correspond bijectively
to continuous complex characters of
\(\operatorname{Gal}(\mathbf Q^{\mathrm{ab}}/\mathbf Q)\), and to
finite-order Hecke characters of \(\mathbf Q\), by \[
\rho_\chi(\sigma_a)=\chi(a),\qquad
\omega_\chi(t,u)=\chi(u^{-1}).
\tag{8}
\] The ideal character at an unramified prime is
\(\omega_\chi(\varpi_p)=\chi(p)\). Their finite conductor exponents
agree. If \(L/\mathbf Q\) is finite abelian and \(X(L)\) is its
character group inflated to (7), then \[
f_L=\operatorname{lcm}_{\chi\in X(L)}f(\chi)
\tag{9}
\] is its finite conductor and its least cyclotomic modulus. Its full
global conductor includes \(\infty\) exactly when \(L\) is not totally
real.

\textbf{Proof.} A continuous character from a profinite group to
\(\mathbf C^\times\) has open kernel. To check this, choose a small
neighborhood of \(1\) containing no nontrivial subgroup, for example
\(1/2<|z|<2\), \(|\arg z|<\pi/2\). All powers force a subgroup element's
modulus to be \(1\), and a nonidentity circle element has a power
outside that arc. A sufficiently small open subgroup of the profinite
domain maps into this neighborhood and hence trivially. The resulting
quotient is finite. Applying this to (7) shows that every character
factors through some \(U_m\).

A finite-order Hecke character kills \(\mathbf R_{>0}\), since a
connected group's continuous image in a finite group is trivial.
Conversely a finite-image character on the unit factor in (4) defines
such a Hecke character. Formula (6) gives exactly the inverse in (8),
and gives \(\chi(p)\) at \(p\nmid f(\chi)\). Its real sign is
\(\chi(-1)\).

For the conductor, the Chinese remainder decomposition of \(U_m\) gives
a least exponent in each prime-power factor through which a character
factors. These are the least exponents killing the local principal-unit
groups; inversion changes neither the kernel nor the exponent. This
proves agreement with local and Hecke conductors. The finite character
lemma proved before section 4 of
\href{../hilberts-theorem-90-and-kummer-theory.html}{Hilbert's theorem
90 and Kummer theory} says that the characters of
\(\operatorname{Gal}(L/\mathbf Q)\) separate its elements. Thus a local
unit subgroup kills the field exactly when it kills every character. Its
least exponent is their maximum, proving (9).

Equivalently \(L\subseteq\mathbf Q(\zeta_m)\) precisely when every
character factors modulo \(m\), by the annihilator description of a
fixed field. The least local exponents show that this is equivalent to
\(f_L\mid m\), proving minimality without assuming a chosen ambient
modulus. Complex conjugation is \(-1\) under (7). It is trivial on the
field exactly when every \(\chi\in X(L)\) is even, which is precisely
that \(L\) is totally real. The real conductor statement follows.
\(\square\)

Multiplication in \(X(L)\) means multiplication after inflation to a
common modulus, followed by taking the primitive representative.
Multiplying zero-extended arithmetic functions would impose an incorrect
conductor on the result.

\subsection{5. The global conductor--discriminant
formula}\label{the-global-conductordiscriminant-formula}

\textbf{Theorem 20.6.} For a finite abelian number field
\(L/\mathbf Q\), \[
|d_L|=\prod_{\chi\in X(L)}f(\chi).
\tag{10}
\] The trivial character contributes \(1\). These are finite conductors;
an infinity factor is not an integer factor of the discriminant.

\textbf{Proof.} Fix a rational prime \(p\), and let \(D\) be its
decomposition group in \(G=\operatorname{Gal}(L/\mathbf Q)\). Its
completion \(E/\mathbf Q_p\) has group \(D\), and there are \(g=[G:D]\)
primes above \(p\). Restriction \(\widehat G\to\widehat D\) is onto and
each fiber has size \(g\): a character of a subgroup of a finite abelian
group extends by successively choosing a root for the value of a newly
adjoined generator. Its kernel is \(\widehat{G/D}\). The local conductor
is computed by restriction to \(D\). Theorem 10.5 therefore gives \[
\sum_{\chi\in X(L)}v_p(f(\chi))
=g\sum_{\eta\in\widehat D}a(\eta)
=g\,v_p(\mathfrak d_{E/\mathbf Q_p}).
\tag{11}
\]

This is also \(v_p(d_L)\). To verify the local/global identification
directly, tensor an integral basis and its trace matrix with
\(\mathbf Z_p\). The Chinese remainder theorem on the quotients by
\(p^n\), followed by completion, gives \[
\mathcal O_L\otimes\mathbf Z_p\simeq
\prod_{\mathfrak P\mid p}\mathcal O_{L_{\mathfrak P}}.
\] Trace on this product is the sum of the local traces. Choosing a
basis in each factor makes the trace pairing block diagonal, so its
determinant valuation is the sum of the local discriminant valuations.
Integral changes of \(\mathbf Z_p\)-basis have unit determinant and do
not alter that valuation. In the Galois case all \(g\) local factors are
isomorphic, giving the last term of (11). Equality of prime exponents
proves (10); absolute value removes the archimedean signature sign.
\(\square\)

For \(L=\mathbf Q(\zeta_5)\), its four characters have conductors
\(1,5,5,5\), so \(|d_L|=125\). For \(L=\mathbf Q(\zeta_{12})\), the four
characters are \(1,\chi_{-3},\chi_{-4},\chi_{12}\), with conductors
\(1,3,4,12\); their product is \(144\). Both agree with the
discriminants calculated independently in the cyclotomic-fields lesson.

\subsection{6. Cyclotomic values and the adelic component
space}\label{cyclotomic-values-and-the-adelic-component-space}

Put \(e(r)=\exp(2\pi ir)\) when it denotes a complex number. In the
Bost--Connes algebra the same notation labels a diagonal arithmetic
generator; its value in a cooled extremal phase indexed by
\(v\in\widehat{\mathbf Z}^{\times}\) is \(e(vr)\). The construction and
classification of those states are proved in
\href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/proof-dependencies.html\#QSM-L04}{Phase
transition in the Bost--Connes system}, sections 3, 4, 6 and 18. We now
specify the number theory in that formula.

\textbf{Proposition 20.7.} The values \(e(a/N)=\zeta_N^a\), for rational
residues \(a/N\), generate \(\mathbf Q^{\mathrm{ab}}\). For every
compatible unit \(b\), \[
\sigma_b\bigl(e(a/N)\bigr)=e(ba/N),\qquad
\sigma_b\bigl(e(va/N)\bigr)=e(bva/N).
\tag{12}
\] The adelic component space used for the cyclotomic tower has a
canonical topological identification \[
\mathbf Q^\times\backslash
\bigl(\mathbf A_f^\times\times\{\pm1\}\bigr)
\simeq \mathbf A_f^\times/\mathbf Q_{>0}^\times
\simeq\widehat{\mathbf Z}^{\times}.
\tag{13}
\] Here a rational \(q\) acts by
\((a,\epsilon)\mapsto(qa,\operatorname{sgn}(q)\epsilon)\). It is the
zero-dimensional space denoted
\(\operatorname{Sh}(\mathrm{GL}_1,\{\pm1\})\) in the Bost--Connes
discussion. With the geometric reciprocity convention, a normalized unit
idèle \(b\) gives \(\sigma_b\); its action on cyclotomic embedding
labels in (13) is multiplication by \(b\).

\textbf{Proof.} The values contain \(\zeta_N=e(1/N)\) for every \(N\),
and every displayed value is a root of unity. Thus their generated field
is the cyclotomic union, equal to (3). Equations (12) are exactly
exponent multiplication in (1), including after replacing \(a/N\) by a
residue indexed by \(v\).

For (13), every orbit has a representative with sign \(+1\), obtained by
multiplication by \(-1\) when necessary. Two such representatives differ
by a positive rational. For \(a\in\mathbf A_f^\times\), multiplication
by \(q=\prod_p p^{-v_p(a_p)}\) produces a unit tuple. A positive
rational unit everywhere is \(1\), so that tuple is unique. The
normalization is continuous on neighborhoods with fixed valuations, and
its inverse is the inclusion of the compact unit tuples. This proves the
topological identifications, not just a set bijection. The unit group is
totally disconnected, so every connected component of this space is a
point.

At level \(N\), reduction of its unit coordinate is the label
\(v\in U_N\) of the embedding \(\zeta_N\mapsto\zeta_N^v\). This
identifies its inverse system with the complex points of the cyclotomic
tower \(\operatorname{Spec}\mathbf Q(\zeta_N)\). Composition with
\(\sigma_b\) sends that embedding to the one labeled \(bv\), proving the
Galois action. Deligne's chosen reciprocity sends a uniformizer to
geometric Frobenius, so it is \(\operatorname{rec}_{\mathrm{geom}}\).
Formula (6) identifies its unit coordinate with precisely \(\sigma_b\),
as asserted. \(\square\)

For the arithmetic normalization the same \(\sigma_b\) is the image of
the unit idèle \(b^{-1}\). Thus (12) uses the cyclotomic exponent \(b\);
substituting an arithmetic idèle coordinate for that exponent requires
the inverse. The arithmetic symmetry formulas in
\href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/proof-dependencies.html\#QSM-L03}{The
Bost--Connes Hecke algebra}, section 6, use the exponent coordinate.
Their operator algebra and state proofs remain in those lessons.

Hilbert's twelfth problem asks for explicit generators of abelian
extensions through special values of functions. Roots of unity solve it
over \(\mathbf Q\) by (3). Imaginary quadratic fields motivate elliptic
functions and complex multiplication; their general generation theory is
beyond this lesson. The preceding lesson obtains its particular ring
class fields by independently checked radical equations.

\subsection{7. Exercises and complete
solutions}\label{exercises-and-complete-solutions}

\subsubsection{Exercise 1 --- The least cyclotomic modulus
(easy)}\label{exercise-1-the-least-cyclotomic-modulus-easy}

Find the smallest \(m\) containing \(\mathbf Q(\sqrt{-7})\) and
\(\mathbf Q(\sqrt3)\) in \(\mathbf Q(\zeta_m)\).

\textbf{Solution.} Their fundamental discriminants are \(-7\) and
\(12\). The quadratic conductor calculation in lesson 18 gives finite
conductors \(7\) and \(12\), respectively. Proposition 20.5 proves that
each is its least cyclotomic modulus. Inclusion at \(7\) is also
exhibited by the quadratic Gauss sum \(\sum_{a\bmod7}(a/7)\zeta_7^a\),
whose square is \(-7\), proved in Proposition 12.6 of \emph{Cyclotomic
fields}. At \(12\), \(\zeta_{12}+\zeta_{12}^{-1}=\sqrt3\). The real
factor of the first field's conductor is \(\infty\); it does not change
the finite least modulus.

Similarly \(\sqrt2=\zeta_8+\zeta_8^{-1}\), and \(\sqrt5\) belongs to
\(\mathbf Q(\zeta_5)\) by its quadratic Gauss sum. Their least moduli
are \(8\) and \(5\), as their conductors predict.

\subsubsection{Exercise 2 --- The negative real idèle
(medium)}\label{exercise-2-the-negative-real-iduxe8le-medium}

Compute the Artin image of the idèle equal to \(-1\) at infinity and
\(1\) at every finite prime, in both conventions.

\textbf{Solution.} Its normalized representative under (5) is
\((1,(-1)_p)\). Formula (6) sends it to \(\sigma_{-1}\) in either
convention, because this unit tuple is its own inverse. It sends every
root of unity to its inverse, hence is complex conjugation. This idèle
is not the diagonal principal element \(-1\): the latter has \(-1\) at
the finite primes too and has trivial global symbol. Their product
displays explicitly the cancellation of the real and finite signs on a
principal idèle.

\subsubsection{Exercise 3 --- The real ray field
(medium)}\label{exercise-3-the-real-ray-field-medium}

Prove that \(\mathbf Q(\zeta_m)^+\) is the ray field for modulus \(m\),
including \(m=1,2\).

\textbf{Solution.} In the ideal ray group, a positive generator
represents the same ideal as its negative. Removing the positivity
condition therefore replaces \(U_m\) by \(U_m/\langle-1\rangle\). The
full ray field for \(m\infty\) is \(\mathbf Q(\zeta_m)\). Passing to
modulus \(m\) kills the image of the real sign, whose Artin symbol is
\(\sigma_{-1}\). The fixed field of that subgroup is exactly the real
subfield. When \(m>2\), this subgroup has order \(2\), so the degree is
\(\varphi(m)/2\). When \(m=1,2\), both groups and fields are trivial,
and taking a half of \(\varphi(m)\) would be incorrect.

\subsubsection{Exercise 4 --- A proof from local Kronecker--Weber
(hard)}\label{exercise-4-a-proof-from-local-kroneckerweber-hard}

Prove the finite assertion in Theorem 20.2 using local Kronecker--Weber
and the absence of a nontrivial abelian extension of \(\mathbf Q\)
unramified at every finite prime.

\textbf{Solution.} Let \(S\) contain the finite primes ramified in
\(L/\mathbf Q\). At \(p\in S\), local Kronecker--Weber puts each
completion of \(L\) in \(\mathbf Q_p(\zeta_{p^{n_p}},\zeta_{b_p})\),
with \(p\nmid b_p\), for a sufficiently large \(n_p\). The second factor
is unramified. Therefore, after adjoining the \(p^{n_p}\)-th roots, the
compositum of the completion with that cyclotomic local field is
unramified over it: it is a subextension of the base change of the
unramified factor.

Put \(m=\prod_{p\in S}p^{n_p}\), \(M=\mathbf Q(\zeta_m)\), and \(T=LM\).
At \(p\in S\), the completion of \(M\) is the product of the totally
ramified \(p\)-primary cyclotomic extension and unramified
prime-to-\(p\) factors. The preceding observation makes \(T/M\)
unramified there. Outside \(S\), both \(L\) and \(M\) are unramified, so
their compositum is too. All these local assertions follow from
separable residue-field extensions and their preservation under base
change.

The group \(G=\operatorname{Gal}(T/\mathbf Q)\) is abelian. Its inertia
subgroup \(I_p\) maps isomorphically onto the inertia of
\(M/\mathbf Q\): restriction is onto, and its kernel is relative inertia
for \(T/M\), already trivial. The cyclotomic inertia theorem gives
\(|I_p|=\varphi(p^{n_p})\). The subgroup \(J\) generated by these finite
inertia groups has fixed field unramified at every finite prime over
\(\mathbf Q\). Its fixed field is abelian, and is therefore
\(\mathbf Q\), since the narrow Hilbert class group of \(\mathbf Q\) is
trivial. Hence \(J=G\). Because \(G\) is abelian, multiplication of the
finitely many inertia subgroups is onto, and \[
|G|\leq\prod_{p\in S}|I_p|=\prod_{p\in S}\varphi(p^{n_p})
=\varphi(m)=[M:\mathbf Q].
\] But \(M\subseteq T\) gives the reverse degree inequality. Thus
\(T=M\), so \(L\subseteq M\), as required. If \(S\) is empty, the narrow
Hilbert argument already gives \(L=\mathbf Q\). This proof uses no
assertion about the class number of \(M\); relative unramifiedness by
itself would not imply \(T=M\).

\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}

The lesson proves both Kronecker--Weber routes, the rational Artin map,
conductors and discriminants. Section 6 proves the cyclotomic values and
the canonical topological component identification. The Bost--Connes
state construction retains its separately identified programme
prerequisite.

\begin{itemize}
\tightlist
\item
  \href{https://www.jmilne.org/math/CourseNotes/ANT.pdf}{J. S. Milne,
  Algebraic Number Theory}.
\item
  \href{https://www.jmilne.org/math/CourseNotes/CFT.pdf}{J. S. Milne,
  Class Field Theory, version 4.03}.
\item
  \href{https://alainconnes.org/wp-content/uploads/bookwebfinal-2.pdf}{Alain
  Connes and Matilde Marcolli, Noncommutative Geometry, Quantum Fields
  and Motives, author-posted draft}.
\end{itemize}

The arithmetic and geometric Frobenius conventions are compared in
equations (6) and (13). The Connes--Marcolli reference is the
author-posted draft, with its own section numbering and pagination.

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

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