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\section{The global reciprocity law}\label{the-global-reciprocity-law}

\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).}

A local symbol tells how one completion acts on an abelian extension.
The global question is whether the symbols at all completions can be
assembled into an invariant of an idèle class. Multiplying them defines
a map on idèles immediately. It does not immediately define a map on
classes: a principal element appears at every place, and its local
contributions must cancel.

We first inspect that cancellation in two explicit examples. The proof
then has two tasks. Construct a map that is already defined on classes,
using the abstract reciprocity theorem and the global cyclic norm axiom.
Identify it with the local product by comparing one local norm quotient
at a time. The identification proves both the principal product law and
the exact global norm kernel.

The abstract theorem is
\href{../the-reciprocity-law-and-the-class-field-correspondence.html}{The
reciprocity law and the class field correspondence}, Theorem 5.1. Its
arithmetic input here is the cyclic axiom in
\href{../the-norm-index-bound-and-hasses-norm-theorem.html}{The norm
index bound and Hasse's norm theorem}, sections 1--8; the later
quadratic-form proof is not used. Local symbols and their explicit
cyclotomic convention were proved in lessons 6 and 9. Fixed idèle
classes and compact norm-one classes were proved in lessons 13 and 14.
The data will use integer degree for function fields and a profinite
degree for number fields, so the valuation hypothesis must allow both.

\subsection{1. The cancellation that the theorem must
explain}\label{the-cancellation-that-the-theorem-must-explain}

Take the principal element \(-6\) in \(\mathbf Q(\zeta_5)/\mathbf Q\).
At 2 and 3 its local symbols act on \(\zeta_5\) with exponents 2 and 3.
At 5 it is a unit, whose exponent is \((-6)^{-1}\equiv4\pmod5\). At
infinity it is negative, whose exponent is \(-1\equiv4\pmod5\). All
other finite components are unramified units and act trivially. The
total exponent is \[
2\cdot3\cdot4\cdot4\equiv1\pmod5.
\] Each ramified or infinite correction matters. Looking only at the
prime divisors of 6 would give exponent 1 modulo 5 in this particular
case, but it would hide two nontrivial compensating actions. The
individual contributions, rather than their coincidental partial
product, are what the theorem must organize.

For a function-field example, take \(K=\mathbf F_7(t)\), a constant
extension of degree six, and \(a=(t^2+1)/t^2\). Since \(-1\) is not a
square in \(\mathbf F_7\), the numerator defines a degree-two place. Its
valuation is 1, while the degree-one place \(t\) has valuation \(-2\),
and infinity has valuation zero. Their constant-field Frobenius
exponents add to \(2\cdot1+1\cdot(-2)=0\). The product is again the
identity. Here cancellation is a degree equation rather than an
archimedean sign correction.

These examples establish the rule only for cyclotomic or constant
extensions, using the explicit local results. For arbitrary abelian
extensions the principal product law is still to be proved. In
particular, we cannot assume a map on \(C_K\) merely because we have
written a finite product on \(J_K\).

\subsection{2. The product on idèles}\label{the-product-on-iduxe8les}

For a finite abelian extension \(L/K\), choose \(w\mid v\). The local
Galois group is the decomposition group
\(D_w\subseteq\operatorname{Gal}(L/K)\). Since the latter is abelian,
changing \(w\) conjugates this subgroup and its map trivially. Define \[
[x,L/K]=\prod_v\operatorname{rec}_{L_w/K_v}(x_v),
\qquad x\in J_K.
\tag{1}
\] At almost all finite places the extension is unramified and \(x_v\)
is a unit, so the factor is \(1\). The arithmetic convention sends a
uniformizer in an unramified extension to residue-field Frobenius
\(z\mapsto z^{Nv}\).

The map is continuous: finitely many local open norm kernels, together
with units outside a finite set, give an open subgroup of its kernel. An
idèle norm lies in the kernel, because each local factor is a norm from
the product of the completions over that place.

\textbf{Proposition 16.1.} Suppose \(K\subseteq K'\), \(L\subseteq L'\),
and both \(L/K\) and \(L'/K'\) are finite abelian. Then \[
[N_{K'/K}x,L/K]
=[x,L'/K']|_L.
\tag{2}
\] The symbols also commute with conjugation and restriction in towers.
Each map (1) is surjective.

\textbf{Proof.} The \(v\)-component of \(N_{K'/K}x\) is
\(\prod_{v'\mid v}N_{K'_{v'}/K_v}x_{v'}\). Local norm functoriality,
proved in \href{../local-reciprocity-and-norm-groups.html}{Local
reciprocity and norm groups}, identifies its symbol with the product of
the restrictions of the \(v'\)-symbols. Multiplication over \(v\) proves
(2). Local conjugation and tower compatibility give the other two
identities.

The image of (1) contains each decomposition group, since local
reciprocity is onto and we may vary just that idèle component. The fixed
field of the image therefore splits completely at every place of \(K\).
Corollary 14.4 says that a nontrivial finite separable extension has
infinitely many places failing complete splitting. The fixed field must
be \(K\), so the image is the whole Galois group. \(\square\)

For an infinite abelian extension, define the symbol by these compatible
finite quotients. This is equivalently a convergent product of local
symbols in the profinite group: in any fixed finite quotient only
finitely many factors are nontrivial. At an infinite local extension we
use the union of its finite localizations, not an assertion that an
algebraic union is complete.

\subsection{3. Principal cyclotomic and constant-field
symbols}\label{principal-cyclotomic-and-constant-field-symbols}

\textbf{Proposition 16.2.} If \(K\) is a number field,
\(a\in K^\times\), and \(\zeta\) is a root of unity, then \[
[a,K(\zeta)/K]=1.
\tag{3}
\] For a function field, every principal idèle likewise has trivial
symbol in every finite constant extension.

\textbf{Proof for number fields.} By (2), restriction of the left side
to \(\mathbf Q(\zeta)\) equals
\([N_{K/\mathbf Q}a,\mathbf Q(\zeta)/\mathbf Q]\). Restriction is
injective on \(\operatorname{Gal}(K(\zeta)/K)\), so it suffices to work
over \(\mathbf Q\). Split the order of \(\zeta\) into prime powers; it
suffices to treat order \(\ell^m\).

Write \(a=u_\ell\ell^{v_\ell(a)}\). The explicit local theorem in
\href{../explicit-local-reciprocity-and-existence.html}{Explicit local
reciprocity and existence}, Corollary 9.5, gives the exponents acting on
\(\zeta_{\ell^m}\): \[
\begin{array}{c|c}
\text{place}&\text{exponent}\\ \hline
p\ne\ell,\ \ p\text{ finite}&p^{v_p(a)}\\
\ell&u_\ell^{-1}\pmod{\ell^m}\\
\infty&\operatorname{sgn}(a).
\end{array}
\tag{4}
\] Negative valuations use inverse powers modulo \(\ell^m\). Since \[
u_\ell=\operatorname{sgn}(a)\prod_{p\ne\ell}p^{v_p(a)},
\] the product of the exponents in (4) is \(1\) in
\((\mathbf Z/\ell^m)^\times\). This proves (3), including the real sign.

\textbf{Proof for function fields.} Let \(k=\mathbf F_q\) be the full
constant field of \(K\). In \(K_m=K\mathbf F_{q^m}\), the local
extension at \(v\) is unramified, with residue degree
\(m/\gcd(m,\deg v)\). Its uniformizer symbol acts on constants as the
\(\deg v\)th power of \(q\)-Frobenius. Thus the product symbol of a
principal element is Frobenius to the power \(\sum_v\deg(v)v(a)=0\), by
the product formula. \(\square\)

Here \([K_m:K]=m\). Indeed an irreducible finite-field polynomial
remains irreducible over \(K\): a monic factor would have coefficients
in the algebraic constant field as well as in \(K\), hence in \(k\),
contradicting irreducibility. Its roots generate the constant extension.
This also shows that the full constant field of \(K_m\) is
\(\mathbf F_{q^m}\).

For example, take \(a=-1\), \(L=\mathbf Q(i)\). At \(2\) the inverse
unit \(-1\) gives complex conjugation, and at infinity the negative sign
gives the same automorphism. Every odd place contributes \(1\). The two
conjugations multiply to \(1\).

\subsection{4. Constructing a map on idèle
classes}\label{constructing-a-map-on-iduxe8le-classes}

The special cases provide exactly the degree direction needed for a map
on classes. The remaining input is the cyclic norm axiom, already proved
without global reciprocity. It is useful to view these inputs as a
checklist: fixed classes, a procyclic Galois degree, a compatible
valuation of norm images, and the two cyclic Tate conditions.

\subsubsection{The module and its fixed
classes}\label{the-module-and-its-fixed-classes}

Fix a separable closure of a ground global field \(k_0\), and put \[
A=\varinjlim_{K/k_0\text{ finite separable}}C_K.
\tag{5}
\] The transition maps are the diagonal embeddings. They are injective
by lesson 13. Give \(A\) the discrete topology for its Galois-module
structure; every element is defined over a finite extension, so its
stabilizer is open. This does not change the ordinary locally compact
topology used on each \(C_K\).

One has \(A^{G_K}=C_K\). To check this, represent a fixed element in
\(C_M\) with \(M/K\) finite Galois. Invariance in the direct limit is
invariance in \(C_M\), by injectivity. The invariant descent
\(C_M^{\operatorname{Gal}(M/K)}=C_K\) was proved using Hilbert 90 in
lesson 13. The group-theoretic module norm is exactly the idèle-class
norm, since both are the product of Galois conjugates. Thus the cyclic
class field axiom of Theorem 15.1 is the precise axiom required by the
abstract theorem.

What remains is its degree and valuation data.

\subsubsection{Integer degree for function
fields}\label{integer-degree-for-function-fields}

For a function field \(k_0\) with full constants \(\mathbf F_q\), let
\(\widetilde{k_0}=k_0\overline{\mathbf F}_q\). Arithmetic constant-field
Frobenius supplies \(d:G_{k_0}\to\widehat{\mathbf Z}\). For a finite
separable \(K/k_0\), \(f_K\) is the degree of its full constant field
over \(\mathbf F_q\), and \(d_K\) is Frobenius normalized for those full
constants.

The degree map \[
v_K(c)=\deg_K c=\sum_v[\kappa(v):k_K]v(x_v),
\qquad c=[x]\in C_K,
\tag{8}
\] is well-defined by the product formula. It is onto \(\mathbf Z\). To
prove the last assertion without a point-count estimate, let \(\delta\)
be the greatest common divisor of all place degrees. If a prime
\(\ell\mid\delta\), every residue field contains the degree-\(\ell\)
constant field. The nontrivial constant extension
\(K\mathbf F_{|k_K|^\ell}/K\) then splits completely at every place,
contrary to Corollary 14.4. Hence \(\delta=1\). Bézout applied to
finitely many of the degrees produces an idèle of degree \(1\).

For every finite separable \(L/K\), the local valuation formula for
norms gives \[
\deg_K(N_{L/K}c)=[k_L:k_K]\deg_L(c).
\tag{9}
\] Indeed the coefficient at \(v\) is the sum of
\([\kappa(v):k_K]f(w/v)v_w(x_w)
=[k_L:k_K][\kappa(w):k_L]v_w(x_w)\). Here \([k_L:k_K]=f_{L/K}\), the
index of the constant-field degree images. Thus the henselian conditions
hold with \(V=\mathbf Z\subset\widehat{\mathbf Z}\); surjectivity means
onto \(\mathbf Z\), not onto its profinite completion. By Proposition
16.2 the constant-extension symbol is precisely Frobenius to this
integer degree.

\subsubsection{Profinite degree for number
fields}\label{profinite-degree-for-number-fields}

Take \(k_0=\mathbf Q\). In lesson 1 we proved \[
\operatorname{Gal}(\mathbf Q(\mu_\infty)/\mathbf Q)
\simeq\widehat{\mathbf Z}^{\,\times}
\simeq\widehat{\mathbf Z}\times T,
\] where \(T\) is a product of finite torsion factors. The closure of
the finite-order elements is \(T\). It matters that we take the closure:
the product \(T\) can contain elements of infinite order. Its fixed
field \(\widetilde{\mathbf Q}\) has Galois group
\(\widehat{\mathbf Z}\). Choose an isomorphism once, obtaining
\(d:G_{\mathbf Q}\to\widehat{\mathbf Z}\).

For a number field \(K\), set \[
f_K=[K\cap\widetilde{\mathbf Q}:\mathbf Q],
\qquad d_K=f_K^{-1}d|_{G_K},
\qquad\widetilde K=K\widetilde{\mathbf Q}.
\] Proposition 16.2 lets the product symbol in \(\widetilde K/K\)
descend to \(C_K\). Define \[
v_K(c)=d_K[c,\widetilde K/K]\in\widehat{\mathbf Z}.
\tag{6}
\] Here \(d_K\) denotes its induced isomorphism on that procyclic Galois
group.

This map is onto. Proposition 16.1 makes its image onto every finite
quotient, hence dense. Split \(C_K=C_K^1\times\mathbf R_{>0}\) using a
chosen archimedean component. The positive-real factor is divisible and
every finite quotient kills it: take an \([L:K]\)th root before applying
a finite symbol. The image consequently comes from compact \(C_K^1\), is
closed, and is all of \(\widehat{\mathbf Z}\).

For every finite extension \(L/K\), norm functoriality gives \[
v_K(N_{L/K}c)=f_{L/K}v_L(c),
\qquad
f_{L/K}=[d(G_K):d(G_L)].
\tag{7}
\] Since \(v_L\) is onto, the norm image under \(v_K\) is exactly
\(f_{L/K}\widehat{\mathbf Z}\). The normalization also gives
\(v_K=f_K^{-1}v_{\mathbf Q}N_{K/\mathbf Q}\). These are exactly the
henselian valuation conditions of lesson 4, with value group
\(V=\widehat{\mathbf Z}\); an integer-only condition would be
inappropriate here.

\textbf{Proposition 16.3.} The Galois module (5), the cyclotomic degree
and valuation (6) for number fields, or the constant-field degree and
valuation (8) for function fields, form a class field theory in the
sense of lessons 4--5.

\textbf{Proof.} The invariant-module and norm identifications are those
just proved for (5). The degree maps are continuous surjections, with
normalized images for finite subfields as in lesson 4. The two valuation
constructions above prove the henselian norm-image conditions. Both
value groups satisfy \(V/nV\simeq\mathbf Z/n\). The cyclic axiom is
Theorem 15.1. These verify every input to Theorem 5.1. \(\square\)

\subsection{5. Comparing the maps on local norm
quotients}\label{comparing-the-maps-on-local-norm-quotients}

The map on classes supplied by the abstract theorem has the correct norm
kernel, but that does not yet identify the action of a single
completion. The local product is the map whose completion-by-completion
behavior we want. For both maps a local norm dies, so the comparison
reduces to a finite quotient. The following proof makes every class of
that quotient accessible from an extension on which the comparison is
already known.

Apply the abstract theorem. For every finite Galois \(L/K\) it gives \[
r_{L/K}:\operatorname{Gal}(L/K)^{\mathrm{ab}}
\xrightarrow{\sim}C_K/N_{L/K}C_L.
\tag{10}
\] Write \(\rho_{L/K}\) for its inverse on \(C_K\). It is continuous
because its kernel is the open norm subgroup. Norm, conjugation and
tower compatibility are Proposition 5.2.

For finite subextensions of \(\widetilde K/K\), its formula is \[
\rho_{L/K}(c)=\varphi_{L/K}^{\,v_K(c)\bmod[L:K]}.
\tag{11}
\] By the definition of \(v_K\), this equals \([c,L/K]\). We must extend
the comparison to every abelian extension.

\subsubsection{Making a local quotient accessible from a degree
extension}\label{making-a-local-quotient-accessible-from-a-degree-extension}

At a finite place of a number field, every cyclotomic
\(\mathbf Z_p\)-extension has infinite local degree. If the residue
characteristic is \(p\), the \(p\)-power cyclotomic local degrees grow
without bound by the totally ramified division fields of lesson 8;
adjoining the fixed finite completion cannot bound them. If the residue
characteristic is \(\ell\ne p\), the extension is unramified and its
Frobenius projects to the \(p\)-adic unit \(\ell^{f_v}\) modulo torsion.
This is not torsion, since no positive power of the positive rational
integer \(\ell^{f_v}>1\) equals \(1\). Its closed cyclic image in
\(\mathbf Z_p\) is therefore a nonzero open subgroup. These observations
also apply when \(p=2\), using the finite sign factor.

For function fields the constant \(\mathbf Z_p\)-extension has local
degree image \(\deg(v)\mathbf Z_p\), also nonzero and open. Thus in
either case there is a \(\mathbf Z_p\)-subextension \(T/K\) of
\(\widetilde K/K\) with infinite local degree at the specified finite
place.

Suppose now that \(L/K\) is cyclic of \(p\)-power degree, generated by
\(\sigma\), and its decomposition group at the specified place is the
whole group. Put \(M=LT\), and let \(D\subseteq\operatorname{Gal}(M/K)\)
be the decomposition subgroup there. It maps onto
\(\operatorname{Gal}(L/K)\), and its image in
\(\operatorname{Gal}(T/K)=\mathbf Z_p\) is open. The image of the kernel
of \(D\to\operatorname{Gal}(L/K)\) is still open, since that kernel has
finite index in \(D\).

Choose \(g\in D\) restricting to \(\sigma\), and adjust it by this
kernel so its \(\mathbf Z_p\)-coordinate is a nonzero positive ordinary
integer. This is possible because the allowed coordinates form a coset
of an open subgroup \(p^b\mathbf Z_p\), and every such coset contains a
positive integer. Let \(H=\overline{\langle g\rangle}\), and \(K'=M^H\).
The coordinate map on \(H\) is injective, with open image, so
\(H\simeq\mathbf Z_p\) and \([K':K]<\infty\). Also
\(H\cap\operatorname{Gal}(M/T)=1\), whence \[
M=K'T,\qquad L'=K'L\subseteq\widetilde{K'}.
\tag{12}
\] Because \(H\subseteq D\), the chosen place has full decomposition
group in \(M/K'\). The finite local group for \(L'/K'\) maps onto
\(\operatorname{Gal}(L/K)\), sending the restriction of \(g\) to
\(\sigma\). This proves the required auxiliary construction, including
its local property.

\subsubsection{Comparing at finite
places}\label{comparing-at-finite-places}

For abelian \(L/K\), compose \(K_v^\times\to C_K\) with \(\rho_{L/K}\),
and compare it with local reciprocity followed by
\(D_v\hookrightarrow\operatorname{Gal}(L/K)\). Both maps kill the local
norm group: a local norm gives a global idèle norm supported at that
place. They therefore factor through the finite local norm quotient
\(D_v\). It suffices to compare on its prime-power elements.

Take such an element \(\sigma\), and replace the base by
\(K_0=L^{\langle\sigma\rangle}\). At the chosen place \(L/K_0\) has full
cyclic decomposition group \(\langle\sigma\rangle\). Apply the auxiliary
construction just proved to get \(L'/K'\) as in (12). Comparison for
\(L'/K'\) is already known by (11), because it is cyclotomic or
constant. Local reciprocity supplies an element \(a'\) whose local
symbol is the lift of \(\sigma\). Norm it down to \(K_{0,v_0}\), then to
\(K_v\). Global norm functoriality for \(\rho\) and local norm
functoriality give \(\sigma\) on both sides. Since both maps already
factor through \(D_v\), this proves their equality. Every element of a
finite abelian group is a product of prime-power elements, so the
comparison holds on all \(K_v^\times\).

\subsubsection{Comparing at infinite
places}\label{comparing-at-infinite-places}

Only a real place becoming complex has nontrivial local group. First
take the special extension \(F(i)/F\) at a real place of a number field
\(F\). Proposition 16.2 makes its product symbol a map on \(C_F\); it
kills norms and is onto, since a negative component at that place maps
to conjugation. Its norm quotient has order \(2\) by Theorem 15.1. There
is just one isomorphism between groups of order \(2\), so this product
map is \(\rho_{F(i)/F}\).

For a general complexification, let \(\sigma\) be its conjugation
element and pass to \(K_0=L^{\langle\sigma\rangle}\). In \(M=L(i)\),
extend \(\sigma\) by the chosen complex conjugation, and let \(K'\) be
its fixed field. The chosen completion of \(K'\) is real, and \[
L'=K'L=M=K'(i).
\] The equality follows since both \(L\) and \(i\) are moved by that
conjugation and \(M/K'\) has degree \(2\). Norm functoriality now
reduces the comparison to the special case, exactly as above. At real
places staying real and at complex places both maps are trivial because
the local norm group is all of \(K_v^\times\).

\subsection{6. Artin reciprocity}\label{artin-reciprocity}

\textbf{Theorem 16.4 (Artin reciprocity).} For every finite abelian
extension \(L/K\) of global fields, the product (1) induces an
isomorphism \[
C_K/N_{L/K}C_L\xrightarrow{\sim}\operatorname{Gal}(L/K).
\tag{13}
\] In particular \[
\prod_v\operatorname{rec}_{L_w/K_v}(a)=1
\quad(a\in K^\times),
\tag{14}
\] and \[
J_K/(K^\times N_{L/K}J_L)\simeq\operatorname{Gal}(L/K).
\tag{15}
\]

\textbf{Proof.} Section 5 identifies the product and \(\rho\) on every
idèle supported at one place. Finite products of such idèles are dense
in the restricted product: a basic neighborhood restricts only finitely
many components, with unit conditions on the tail. Both maps are
continuous to a finite group, so they agree everywhere on \(J_K\). The
abstract map factors through \(C_K\), proving (14), and has kernel
\(NC_L\), proving (13)--(15). \(\square\)

For an unramified finite place, the global symbol of its idèle
uniformizer is its arithmetic Frobenius. Thus this theorem includes the
usual Frobenius prescription at primes, as well as every ramified and
archimedean component. For nonabelian finite Galois extensions (10)
remains valid, and the norm group equals that of the maximal abelian
subextension by norm limitation in lesson 5.

For an infinite abelian extension the map on \(C_K\) has dense image,
because it is onto every finite quotient. Over number fields it is onto:
the positive-real factor is killed and the image of compact \(C_K^1\) is
closed. Over function fields its constant-field projection is the
integer degree subgroup of \(\widehat{\mathbf Z}\), so it is not onto
the infinite constant extension. The next lesson describes the precise
topology and the existence correspondence.

The construction of the abstract number-field degree involved a choice
of an isomorphism with \(\widehat{\mathbf Z}\). The final map is
independent of that choice, since the comparison identifies it with the
product of the fixed arithmetic local maps.

\subsection{7. Quadratic reciprocity as a product
formula}\label{quadratic-reciprocity-as-a-product-formula}

Let \(p,q\) be distinct odd rational primes, and put
\(p^*=(-1)^{(p-1)/2}p\). Take \(L=\mathbf Q(\sqrt{p^*})\) and the
principal element \(q\). By the explicit Hilbert symbol formulas of
\href{../hilbert-symbols-and-local-conics.html}{Hilbert symbols and
local conics}, its local actions on \(\sqrt{p^*}\) are \[
\begin{array}{c|c}
v&\text{multiplying sign}\\ \hline
q&(p^*/q)\\
p&(q/p)\\
2&1\\
\infty&1.
\end{array}
\tag{16}
\] At \(q\) the extension is unramified and Frobenius gives the residue
square character. At \(p\), the odd-place Hilbert formula has valuations
\(0\) and \(1\), giving \((q/p)\). At \(2\), both entries are odd and
\(p^*\equiv1\pmod4\), so the odd-unit \(2\)-adic Hilbert formula gives
\(1\). At infinity \(q>0\). All other places have an unramified unit
symbol.

Theorem 16.4 now gives \((p^*/q)(q/p)=1\). Since
\((-1/q)=(-1)^{(q-1)/2}\), this is \[
(p/q)(q/p)=(-1)^{(p-1)(q-1)/4}.
\tag{17}
\] No prior global quadratic reciprocity theorem was used: only the
already proved local Hilbert formulas and the global product theorem.

\subsection{8. Exercises and complete
solutions}\label{exercises-and-complete-solutions}

\subsubsection{Exercise 1 --- The real sign
(easy)}\label{exercise-1-the-real-sign-easy}

Verify the product formula for \(-1\) in \(\mathbf Q(i)/\mathbf Q\).

\textbf{Solution.} At \(2\), Corollary 9.5 acts on \(i\) by the inverse
of the unit \(-1\) modulo \(4\), hence sends \(i\) to \(-i\). At
infinity, \(-1\) gives complex conjugation. At every odd prime the
extension is unramified or split and \(-1\) is a unit, so the symbol is
trivial. The product is conjugation squared, hence \(1\). Omitting
infinity would give the wrong product.

\subsubsection{Exercise 2 --- A fifth root of unity
(medium)}\label{exercise-2-a-fifth-root-of-unity-medium}

Verify the product formula for \(3\) in
\(\mathbf Q(\zeta_5)/\mathbf Q\).

\textbf{Solution.} At \(3\), arithmetic Frobenius sends \(\zeta_5\) to
\(\zeta_5^3\). At \(5\), the unit \(3\) acts through its inverse modulo
\(5\), which is \(2\). The positive real component is trivial, and every
other finite component is an unramified unit. Composing the two
nontrivial actions gives exponent \(3\cdot2=6\equiv1\pmod5\).

\subsubsection{Exercise 3 --- The quadratic law
(medium)}\label{exercise-3-the-quadratic-law-medium}

Derive quadratic reciprocity for distinct odd primes from Theorem 16.4.

\textbf{Solution.} Use \(L=\mathbf Q(\sqrt{p^*})\), \(a=q\), and the
four local calculations in (16). The product is \(1\), so
\((p^*/q)=(q/p)\). Expanding \(p^*=(-1)^{(p-1)/2}p\) and evaluating the
finite-field character of \(-1\) gives (17). The special choice
\(p^*\equiv1\pmod4\) makes the \(2\)-adic contribution trivial; using
\(p\) without that sign adjustment would require retaining it.

\subsubsection{Exercise 4 --- Verify the class field theory data
(hard)}\label{exercise-4-verify-the-class-field-theory-data-hard}

Prove Proposition 16.3, including its value-group distinction.

\textbf{Solution.} Injectivity and invariant descent from lesson 13
identify the discrete union module's \(G_K\)-invariants with \(C_K\),
and the coset norm with the idèle norm. Theorem 15.1 supplies both
cyclic Tate groups required by the abstract axiom.

For number fields, take the torsion-closure quotient of the cyclotomic
Galois group as the degree map. Proposition 16.2 kills principal
elements in its product symbol, so (6) is defined on \(C_K\).
Proposition 16.1 makes its image dense in \(\widehat{\mathbf Z}\). The
divisible positive-real factor maps trivially to every finite quotient,
and compact \(C_K^1\) has closed image; the map is onto. Norm
compatibility gives precisely \(f_{L/K}\widehat{\mathbf Z}\) as the
valuation of the norm image.

For function fields, take constant Frobenius as the degree map, and use
integer idèle degree as valuation. If its image had greatest common
divisor \(\delta>1\), a prime dividing \(\delta\) would give a
nontrivial constant extension split at every place, contradicting lesson
14. Thus the image is \(\mathbf Z\). Local norm valuations yield (9),
with \(f_{L/K}\) the full-constant-field degree. Both \(\mathbf Z\) and
\(\widehat{\mathbf Z}\) have quotient \(\mathbf Z/n\) by their
\(n\)-multiples. These verify all hypotheses; falsely requiring the
function-field degree to be onto \(\widehat{\mathbf Z}\), or the
number-field valuation to be integer-valued, would not.

\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 principal cancellation, local comparison and auxiliary procyclic
construction are proved here. Number-field cyclotomic degree and
function-field constant degree are constructed separately; the permitted
profinite valuation image is retained.

\begin{itemize}
\tightlist
\item
  \href{https://www.mathi.uni-heidelberg.de/~schmidt/Neukirch-en/Neukirch_cft_02_may15.pdf}{Jürgen
  Neukirch, Class Field Theory --- The Bonn Lectures, Online Edition 2.0
  (May 2015), edited by Alexander Schmidt}.
\item
  \href{https://kskedlaya.org/cft/sec_abstractcft1.html}{Kiran S.
  Kedlaya, Notes on class field theory, author-hosted HTML edition}.
\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}