Class Field Theory (Overview)¶

Job: classify abelian extensions of global and local fields by arithmetic data (ideals / idèles / norms) via the Artin reciprocity map.

Core¶

Abelian extensions¶

Let $K$ be a number field. A finite extension $L/K$ is abelian if $\mathrm{Gal}(L/K)$ is abelian. Maximal abelian $K^{\mathrm{ab}}/K$ is infinite; class field theory describes $\mathrm{Gal}(K^{\mathrm{ab}}/K)$ and the finite abelian layers by open finite-index subgroups of a class group built from $K$ alone.

Classical special cases.

Extension type	Galois group controlled by
Cyclotomic $\mathbb{Q}(\zeta_m)/\mathbb{Q}$	$(\mathbb{Z}/m\mathbb{Z})^\times$
Ray class fields of $K$	ray class groups $\mathrm{Cl}_{\mathfrak{m}}$
Hilbert class field $H/K$	$\mathrm{Cl}_K$ (maximal unramified abelian)
Local abelian $L/K_v$	$K_v^\times/N_{L/K_v}L^\times$

Ideal-theoretic global CFT (classical form)¶

Fix a modulus $\mathfrak{m}$ (integral ideal + real places). The ray class group $\mathrm{Cl}_{\mathfrak{m}}$ is fractional ideals coprime to $\mathfrak{m}$, modulo principals generated by elements $\equiv 1\bmod\mathfrak{m}$ (positive at real places in $\mathfrak{m}$).

Theorem (existence + Artin reciprocity, classical). For each $\mathfrak{m}$ there is a unique abelian extension $K_{\mathfrak{m}}/K$ (ray class field) with: 1. primes ramifying in $K_{\mathfrak{m}}/K$ divide $\mathfrak{m}$ (archimedean places in $\mathfrak{m}$ may ramify); 2. the Artin map $$ \psi_{K_{\mathfrak{m}}/K}:\mathrm{Cl}{\mathfrak{m}}\xrightarrow{\sim}\mathrm{Gal}(K/K), $$ sends unramified prime ideals }$\mathfrak{p}\nmid\mathfrak{m}$ to the Frobenius $\mathrm{Frob}_{\mathfrak{p}}$.

Hilbert class field: $\mathfrak{m}=1$, $H=K_{(1)}$, $\mathrm{Gal}(H/K)\cong\mathrm{Cl}_K$, unramified at all finite (and infinite) places; every ideal of $\mathcal{O}_K$ becomes principal in $\mathcal{O}_H$ (principal ideal theorem).

Conductor–discriminant: the conductor $\mathfrak{f}_{L/K}$ is the minimal $\mathfrak{m}$ with $L\subset K_{\mathfrak{m}}$; Artin map factors through $\mathrm{Cl}_{\mathfrak{f}}$.

Idèle-theoretic form (modern)¶

Idèles $\mathbb{A}_K^\times=$ restricted product of $K_v^\times$ w.r.t. $\mathcal{O}_v^\times$. Idèle class group $C_K=\mathbb{A}_K^\times/K^\times$.

Theorem (global CFT). There is a continuous Artin reciprocity homomorphism $$ \theta_K:C_K\to\mathrm{Gal}(K^{\mathrm{ab}}/K) $$ with dense image, kernel the connected component $\overline{K_\infty^\times}$ of the identity in the archimedean part (more precisely: $\theta_K$ induces $$ C_K/C_K^\circ\xrightarrow{\sim}\mathrm{Gal}(K^{\mathrm{ab}}/K) $$ in the number field case). Finite abelian $L/K$ correspond to open finite-index subgroups $N_{L/K}C_L\subset C_K$ with $$ C_K/N_{L/K}C_L\cong\mathrm{Gal}(L/K). $$

Compatibility. For unramified $\mathfrak{p}$, the class of a uniformizer in $K_{\mathfrak{p}}^\times\subset\mathbb{A}_K^\times$ maps to $\mathrm{Frob}_{\mathfrak{p}}$.

Local class field theory¶

For a nonarchimedean local field $K$ (finite extension of $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$): $$ \theta_K:K^\times\to\mathrm{Gal}(K^{\mathrm{ab}}/K) $$ is injective with dense image; for finite abelian $L/K$, $$ K^\times/N_{L/K}L^\times\cong\mathrm{Gal}(L/K). $$ Unramified extension of degree $f$ $\leftrightarrow$ subgroup $\langle\varpi\rangle\times\mathcal{O}_K^\times$ norms, Frobenius $\leftrightarrow$ uniformizer. Local–global compatibility: global Artin map restricts to local Artin maps at each place.

Local–global dictionary (overview)¶

Local	Global
$K_v^\times/N L_w^\times$	decomposition / inertia of $v$ in $L/K$
Norm residue symbol	Artin symbol $(\cdot,L/K)$
Local conductor $f_v$	global conductor $\prod_v\mathfrak{p}_v^{f_v}$
Local Kronecker–Weber ($K=\mathbb{Q}_p$)	global Kronecker–Weber ($\mathbb{Q}^{\mathrm{ab}}=\bigcup\mathbb{Q}(\zeta_m)$)

Chebotarev density theorem (beyond abelian, but used constantly): Frobenius conjugacy classes equidistribute by class size; in the abelian case this is Artin map + Dirichlet density on ray classes.

See it¶

Quadratic reciprocity via CFT. For odd primes $p\neq q$, $\bigl(\frac{q}{p}\bigr)=\bigl(\frac{p^*}{q}\bigr)$ follows from $\mathrm{Gal}(\mathbb{Q}(\sqrt{p^*})/\mathbb{Q})\cong\mathrm{Cl}_{\mathfrak{m}}$ description / Artin symbol at $q$ equaling the Kronecker symbol — CFT package specializes to elementary reciprocity laws.

Hilbert class field of $\mathbb{Q}(\sqrt{-5})$. $\mathrm{Cl}\cong C_2$, so $[H:K]=2$. $H=K(\sqrt{-1})=\mathbb{Q}(\sqrt{-5},i)$; the non-principal ideal $(2,1+\sqrt{-5})$ splits/becomes principal according to Artin map generating $\mathrm{Gal}(H/K)$.

Check¶

State the characterizing property of the Hilbert class field $H/K$.

Maximal abelian extension unramified at all places (finite and infinite); equivalently the ray class field of modulus $1$; $\mathrm{Gal}(H/K)\cong\mathrm{Cl}_K$ via $\mathfrak{p}\mapsto\mathrm{Frob}_{\mathfrak{p}}$.

What is the local abelian extension corresponding to the subgroup $N=\langle\varpi\rangle^f\cdot\mathcal{O}_K^\times\subset K^\times$ for nonarchimedean $K$?

The unramified extension of degree $f$: norms from the unramified $L/K$ of degree $f$ are $\langle\varpi^f\rangle\cdot\mathcal{O}_K^\times$ (standard normalization); Artin map sends uniformizer to Frobenius.

Pitfalls¶

Unramified includes infinite places: complex conjugation can ramify in the “infinite” sense; totally real Hilbert class fields need careful infinite modulus.
Idèle vs ideal formulations: ray class groups are quotients of idèle class groups; forgetting archimedean conditions loses infinite ramification data.
Non-abelian extensions are not classified by $C_K$ alone — need non-abelian class field theory / Langlands (Galois representations $\leftrightarrow$ automorphic forms).
Existence theorems are non-constructive in general; explicit class fields beyond CM / cyclotomic / genus theory are hard.

Refs¶

Neukirch, Algebraic Number Theory, Ch. IV–VI
Cassels–Fröhlich, Algebraic Number Theory
Milne, Class Field Theory (notes)
Artin–Tate, Class Field Theory

Next¶

Algebraic NT overview · Foundations · Galois theory · Analytic L-functions · Number theory

Extension type	Galois group controlled by
Cyclotomic \(\mathbb{Q}(\zeta_m)/\mathbb{Q}\)	\((\mathbb{Z}/m\mathbb{Z})^\times\)
Ray class fields of \(K\)	ray class groups \(\mathrm{Cl}_{\mathfrak{m}}\)
Hilbert class field \(H/K\)	\(\mathrm{Cl}_K\) (maximal unramified abelian)
Local abelian \(L/K_v\)	\(K_v^\times/N_{L/K_v}L^\times\)

Local	Global
\(K_v^\times/N L_w^\times\)	decomposition / inertia of \(v\) in \(L/K\)
Norm residue symbol	Artin symbol \((\cdot,L/K)\)
Local conductor \(f_v\)	global conductor \(\prod_v\mathfrak{p}_v^{f_v}\)
Local Kronecker–Weber (\(K=\mathbb{Q}_p\))	global Kronecker–Weber (\(\mathbb{Q}^{\mathrm{ab}}=\bigcup\mathbb{Q}(\zeta_m)\))