Class field theory — abelian extensions and Artin reciprocity

Intuition Beginner

A number field such as $\mathbb{Q}$ or $\mathbb{Q}(\sqrt{-5})$ carries symmetries. Pick the symmetries that commute with each other — where carrying out $a$ then $b$ gives the same result as carrying out $b$ then $a$. These commuting symmetries are called abelian, and the field extensions they govern are abelian extensions. Class field theory is the complete classification of these extensions.

The remarkable claim is a perfect dictionary. Every abelian extension of a number field $K$ matches, one for one, a finite object living inside $K$ itself — a generalised ideal class group. The match sends each abelian extension $L$ over $K$ to a quotient of the class-group data of $K$. So the "nice" symmetries of $K$ are classified by arithmetic already present in $K$.

The simplest case is $\mathbb{Q}$ itself. Every abelian extension of $\mathbb{Q}$ sits inside a field built from roots of unity — a cyclotomic field. This is the Kronecker-Weber theorem, and it is the blueprint for the whole theory. For other number fields the dictionary deepens: the class group itself measures the unramified abelian extensions, and bigger quotients measure ramified ones.

Visual Beginner

The left panel records the arithmetic interior of $K$ — its ideals grouped into ray class groups by congruence conditions. The right panel records the extensions sitting above $K$ — the abelian Galois extensions, with the Hilbert class field $H$ as the largest one in which no prime breaks. The arrow between the panels is the Artin map, and the central theorem is that this arrow is always a perfect matching.

Worked example Beginner

The field $K=\mathbb{Q}(\sqrt{-5})$ has class number $h_K=2$: its ideal class group $\mathrm{Cl}(K)$ has two elements, as computed in 21.17.01. Class field theory predicts that $K$ has a single largest abelian extension $H$ in which no prime ideal of $K$ breaks apart (no prime gets squared when lifted). This $H$ is the Hilbert class field, and the theory forces its degree over $K$ to equal the class number: $[H:K]=h_K=2$.

So there must be a degree-2 extension of $\mathbb{Q}(\sqrt{-5})$ with no ramified primes. The answer is $H=\mathbb{Q}(\sqrt{-5},i)$, obtained by adjoining the square root of $-1$. The new symmetry sending $i\mapsto -i$ commutes with every symmetry of $K$, so the extension $H/K$ is abelian — exactly as the theory requires.

To verify there is no ramification, compare discriminants. The discriminant of $K$ is $d_K=-20$. The discriminant of $H$ is $400$, which equals $(-20{)}^2=d_K^2$. Since $d_H=d_K^2$, the relative discriminant of $H$ over $K$ is the unit ideal: no prime of $K$ ramifies in $H$. The reciprocity isomorphism $\mathrm{Cl}(K)\cong \mathrm{Gal}(H/K)$ — both groups have two elements — is the first concrete instance of the dictionary.

Check your understanding Beginner

Formal definition Intermediate+

A modulus (or cycle) in a number field $K$ is a formal product $\mathfrak{m}={\mathfrak{m}}_0{\mathfrak{m}}_{\infty }$, where ${\mathfrak{m}}_0$ is a nonzero integral ideal of ${\mathcal{O}}_K$ and ${\mathfrak{m}}_{\infty }$ is a subset of the real embeddings $\sigma :K\hookrightarrow \mathbb{R}$ ^{[Childress Ch. 2]}. The ray modulo $\mathfrak{m}$ is the subgroup of $K^{\times }$ defined by

$$K_{\mathfrak{m},1}=\{x\in K^{\times }:v_{\mathfrak{p}}(x-1)\ge v_{\mathfrak{p}}({\mathfrak{m}}_0)\text{for all}\mathfrak{p}\mid {\mathfrak{m}}_0,\sigma (x)>0\text{for all}\sigma \in {\mathfrak{m}}_{\infty }\},$$

that is, elements congruent to $1$ modulo the finite part and strictly positive at each chosen real place. Two fractional ideals lie in the same ray class when they differ by a principal ideal $(x)$ with $x\in K_{\mathfrak{m},1}$.

The ray class group modulo $\mathfrak{m}$ is the quotient

$${\mathrm{Cl}}_{\mathfrak{m}}(K)=I_K^{\mathfrak{m}}/P_{K,1}^{\mathfrak{m}},$$

where $I_K^{\mathfrak{m}}$ is the group of fractional ideals of $K$ coprime to ${\mathfrak{m}}_0$ and $P_{K,1}^{\mathfrak{m}}$ is the subgroup of principal ideals $(x)$ with $x\in K_{\mathfrak{m},1}$ ^{[Neukirch Ch. VI]}. For the empty modulus $\mathfrak{m}=1$ the congruence condition is vacuous and ${\mathrm{Cl}}_1(K)$ recovers the ordinary ideal class group $\mathrm{Cl}(K)$ of 21.17.01. Every ray class group is finite.

Artin symbol (Frobenius). Let $L/K$ be a finite Galois extension and let $\mathfrak{p}$ be a prime of $K$ unramified in $L$. For a prime $\mathfrak{P}\mid \mathfrak{p}$ of $L$, the Artin symbol is the unique element $\left(\frac{L/K}{\mathfrak{p}}\right)\in \mathrm{Gal}(L/K)$ acting on residue classes by

$$\left(\frac{L/K}{\mathfrak{p}}\right)(\bar{\alpha })={\bar{\alpha }}^{N\mathfrak{p}},N\mathfrak{p}=|{\mathcal{O}}_K/\mathfrak{p}|,$$

for every $\bar{\alpha }\in {\mathcal{O}}_L/\mathfrak{P}$. Because $\mathfrak{p}$ is unramified the symbol is independent of the choice of $\mathfrak{P}$ above $\mathfrak{p}$ when $L/K$ is abelian. Extending multiplicatively over factorisations of ideals defines the Artin map

$${\Phi }_{L/K}:I_K^{\mathfrak{m}}⟶\mathrm{Gal}(L/K),\prod _{\mathfrak{p}}{\mathfrak{p}}^{v_{\mathfrak{p}}}\mapsto \prod _{\mathfrak{p}}{\left(\frac{L/K}{\mathfrak{p}}\right)}^{v_{\mathfrak{p}}},$$

defined on the group of ideals coprime to $\mathfrak{m}$, where ${\mathfrak{m}}_0$ is divisible by every finite prime of $K$ that ramifies in $L$ ^{[Artin 1927]}.

The Hilbert class field $H_K$ of $K$ is the maximal abelian extension of $K$ unramified at every finite prime. The narrow Hilbert class field additionally requires unramification at the real places. The ray class field $K_{\mathfrak{m}}$ modulo $\mathfrak{m}$ is the maximal abelian extension of $K$ whose conductor divides $\mathfrak{m}$; equivalently the extension cut out by the Artin map on ${\mathrm{Cl}}_{\mathfrak{m}}(K)$.

Counterexamples to common slips

• Class field theory classifies only abelian extensions. A non-abelian extension such as the splitting field of a generic irreducible quintic with group $S_5$ lies outside the theory; its symmetries do not commute, and the classification passes to the Langlands programme 21.10.01.
• "Unramified" in the Hilbert class field means unramified at every finite prime. At a real place of $K$ the extension may turn it complex; the narrow Hilbert class field forbids this extra behaviour.
• The Artin symbol is defined only at primes unramified in $L/K$. Ramified primes carry no Frobenius, which is why the modulus $\mathfrak{m}$ must absorb every ramified finite prime before the map ${\Phi }_{L/K}$ is defined on $I_K^{\mathfrak{m}}$.

Key theorem with proof Intermediate+

Theorem (Artin reciprocity, global). Let $L/K$ be a finite abelian extension of number fields with Galois group $G=\mathrm{Gal}(L/K)$, and let $\mathfrak{m}$ be a modulus divisible by every finite prime of $K$ that ramifies in $L$. Then the Artin map ${\Phi }_{L/K}:I_K^{\mathfrak{m}}\to G$ is surjective with kernel containing the ray $P_{K,1}^{\mathfrak{m}}$ and the norms $N_{L/K}(I_L^{\mathfrak{m}})$. It descends to an isomorphism

$$I_K^{\mathfrak{m}}/P_{K,1}^{\mathfrak{m}}{N}_{L/K}(I_L^{\mathfrak{m}})\stackrel{\sim }{\to }G.$$

Proof. The argument proceeds in five stages: the Artin map is a homomorphism; its kernel contains the ray-norm subgroup; the first inequality bounds the index from below; surjectivity follows by a group-theoretic reduction; the second inequality bounds the index from above and closes the kernel.

Stage 1 — the Artin map is a homomorphism. For unramified primes $\mathfrak{p},{\mathfrak{p}}^{\prime }$ of $K$, the Frobenius at the product $\mathfrak{p}{\mathfrak{p}}^{\prime }$ is the product of the Frobenii: $\left(\frac{L/K}{\mathfrak{p}{\mathfrak{p}}^{\prime }}\right)=\left(\frac{L/K}{\mathfrak{p}}\right)\left(\frac{L/K}{{\mathfrak{p}}^{\prime }}\right)$. Indeed the residue ring ${\mathcal{O}}_L/\mathfrak{P}$ has $\mathfrak{p}$-Frobenius raising to the power $N\mathfrak{p}$ and ${\mathfrak{p}}^{\prime }$-Frobenius raising to $N{\mathfrak{p}}^{\prime }$, and these commute and compose to the power $N\mathfrak{p}\cdot N{\mathfrak{p}}^{\prime }$ on the separate residue factors. Multiplicative extension over general ideals in $I_K^{\mathfrak{m}}$ gives a well-defined homomorphism.

Stage 2 — the kernel contains the ray and the norms. If $(x)\in P_{K,1}^{\mathfrak{m}}$ with $x\equiv 1(\mathrm{mod}{\mathfrak{m}}_0)$ and $\sigma (x)>0$ for $\sigma \in {\mathfrak{m}}_{\infty }$, then locally at each ramified prime the element $x$ is a unit congruent to $1$, and the local norm-residue computation (the Hilbert symbol of 21.02.05) forces $\left(\frac{L/K}{(x)}\right)$ to act as the identity. Likewise every ideal norm $N_{L/K}(\mathfrak{A})$ lies in the kernel: at an unramified prime $\mathfrak{q}$ of $K$, a prime $\mathfrak{Q}$ of $L$ above it contributes $f(\mathfrak{Q}/\mathfrak{q})$ copies of $\left(\frac{L/K}{\mathfrak{q}}\right)$ to the Artin symbol of $N\mathfrak{A}$, and this contribution is ${\mathrm{Frob}}_{\mathfrak{q}}^{f(\mathfrak{Q}/\mathfrak{q})}=1$ in $\mathrm{Gal}({\mathcal{O}}_L/\mathfrak{Q}/{\mathcal{O}}_K/\mathfrak{q})$ because the residue extension has degree $f$ and Frobenius has order $f$. Hence $P_{K,1}^{\mathfrak{m}}{N}_{L/K}(I_L^{\mathfrak{m}})\subseteq \ker {\Phi }_{L/K}$.

Stage 3 — the first inequality. Let $h_{\mathfrak{m}}(L/K)=(I_K^{\mathfrak{m}}:P_{K,1}^{\mathfrak{m}}{N}_{L/K}(I_L^{\mathfrak{m}}))$. The first inequality (Takagi) asserts $h_{\mathfrak{m}}(L/K)\ge [L:K]$. In the idelic formulation of Chevalley it is a direct count: the kernel of the norm on idele classes $C_L\to C_K$ has index at least $[L:K]$ by the cohomology of the cyclic group $\mathrm{Gal}(L/K)$ acting on $C_L$, the bound falling out of the Herbrand quotient $(C_L)\cong [L:K]$ ^{[Artin-Tate Ch. 7]}. This is the inequality whose proof the modern cohomological machine absorbs.

Stage 4 — surjectivity. The Artin map has image $G^{\prime }=\mathrm{im}({\Phi }_{L/K})\subseteq G$. Let $E=L^{G^{\prime }}$ be its fixed field; then $E\supseteq K$ and ${\Phi }_{L/K}(\mathfrak{p}){|}_E=1$ for every $\mathfrak{p}\in I_K^{\mathfrak{m}}$, so every prime of $K$ splits completely in $E/K$. A prime that splits completely is a norm from $E$, hence $I_K^{\mathfrak{m}}=P_{K,1}^{\mathfrak{m}}{N}_{E/K}(I_E^{\mathfrak{m}})$ and $h_{\mathfrak{m}}(E/K)=1$. If $E\ne K$ the first inequality (Stage 3) applied to $E/K$ gives $1=h_{\mathfrak{m}}(E/K)\ge [E:K]>1$, a contradiction. Hence $E=K$, $G^{\prime }=G$, and ${\Phi }_{L/K}$ is surjective.

Stage 5 — the second inequality and the kernel. The second inequality (Takagi, Chevalley) asserts $h_{\mathfrak{m}}(L/K)\le [L:K]$. Its proof is the analytic-cohomological heart of the theory: in the cohomological form, the Herbrand quotient of $C_L$ under $\mathrm{Gal}(L/K)$ and the vanishing of $H^1$ (Hilbert's theorem 90 for ideles) force the index of the norm subgroup in $C_K$ down to $[L:K]$ ^{[Artin-Tate Ch. 7]}. Combining Stages 3 and 5 yields $h_{\mathfrak{m}}(L/K)=[L:K]=|G|$. Stage 4 gives $G\cong I_K^{\mathfrak{m}}/\ker {\Phi }_{L/K}$ with $P_{K,1}^{\mathfrak{m}}N(I_L^{\mathfrak{m}})\subseteq \ker {\Phi }_{L/K}$, and the two subgroups now have equal finite index, so $\ker {\Phi }_{L/K}=P_{K,1}^{\mathfrak{m}}{N}_{L/K}(I_L^{\mathfrak{m}})$. The induced map $I_K^{\mathfrak{m}}/P_{K,1}^{\mathfrak{m}}N(I_L^{\mathfrak{m}})\stackrel{\sim }{\to }G$ is the reciprocity isomorphism. $\square$

Bridge. Artin reciprocity builds toward the Langlands programme 21.10.01, where the abelian Galois group is replaced by the full non-abelian Galois group and ray class groups are replaced by automorphic representations, and appears again in the Galois-representation theory of 21.05.01, where compatible families of $\ell$-adic representations are the non-abelian shadows of the abelian Artin map. The foundational reason the theory works is that Frobenius elements already form a complete set of symmetries for unramified primes: this is exactly what the Artin map exploits, reading the entire Galois group off residue-field arithmetic. The central insight is that congruence conditions inside $K$ (the ray) and splitting data outside $K$ (the extension) determine each other, the bridge is that this reciprocity is dual to the local norm-residue pairing of 21.02.05, and putting these together the whole theory generalises from the ideal-theoretic form to the idele-theoretic formulation of Chevalley that the modern cohomological proof rests on.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none is recorded for this unit because Mathlib does not yet contain the class-field-theoretic edifice this unit organises. The supporting pillars — the ideal class group ClassGroup, fractional ideals, Galois theory, cyclotomic polynomials, and the Frobenius automorphism at a prime — are present, but there is no definition of a ray class group ${\mathrm{Cl}}_{\mathfrak{m}}(K)$, no construction of a ray class field or the Hilbert class field, no Artin reciprocity map $I_K^{\mathfrak{m}}\to \mathrm{Gal}(L/K)$ with its kernel identification, and no proof that $[H_K:K]=h_K$. The local theory (the local Artin map and the isomorphism $K^{\times }/N_{L/K}{L}^{\times }\cong \mathrm{Gal}(L/K)$) is likewise absent. Closing this gap is a major formalization project — building the first and second inequalities from the cohomology of ideles — rather than local assembly work; the gap note in the unit metadata details what is missing and why it matters.

Advanced results Master

Theorem (Takagi existence theorem, 1920). Let $K$ be a number field. Every congruence subgroup $H$ with $P_{K,1}^{\mathfrak{m}}\subseteq H\subseteq I_K^{\mathfrak{m}}$ (equivalently, every open subgroup of finite index in the idele class group $C_K$) is the kernel of the Artin map of a unique finite abelian extension $L/K$. The map $L\mapsto \ker {\Phi }_{L/K}$ is a bijection between finite abelian extensions of $K$ (inside a fixed closure) and such congruence subgroups, reversing inclusion ^{[Takagi 1920]}. This is the converse of Artin reciprocity: not only does every abelian extension produce a congruence subgroup via the Artin map, but every congruence subgroup comes from a unique abelian extension. The existence theorem turns reciprocity from a property of given extensions into a complete classification.

Theorem (Kronecker-Weber). Every finite abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field $\mathbb{Q}({\zeta }_n)$ for some $n$. This is the case $K=\mathbb{Q}$ of the existence theorem: the ray class fields of $\mathbb{Q}$ are precisely the cyclotomic fields (and their subfields), so every abelian extension of $\mathbb{Q}$ is cyclotomic. The proof reduces, via the structure theorem for finite abelian groups and induction on degree, to the prime-power case, where a ramification analysis (Exercise 7) forces the extension into the cyclotomic field of its conductor. This is the simplest case from which the whole general theory grew, and the explicit reciprocity map for cyclotomic fields is worked out in the Full proof set.

Theorem (local class field theory). Let $L/K$ be a finite abelian extension of $p$-adic fields. The local Artin map $K^{\times }\to \mathrm{Gal}(L/K)$ is surjective with kernel $N_{L/K}(L^{\times })$, inducing

$$K^{\times }/N_{L/K}(L^{\times })\stackrel{\sim }{\to }\mathrm{Gal}(L/K).$$

The global reciprocity law is obtained by assembling these local maps over all places of $K$ via the idèle class group ^{[Serre 1979 Ch. XIII-XV]}. The local law is cleaner than the global one because $K^{\times }$ is a single explicit group (a uniformiser times units), and the unramified case is fully proved in the Full proof set.

Theorem (Hasse norm theorem). For a cyclic extension $L/K$ of number fields, an element $x\in K^{\times }$ is a global norm from $L$ if and only if it is a local norm in every completion $K_v$, that is, $x\in N_{L_v/K_v}(L_v^{\times })$ for every place $v$ ^{[Hasse 1930]}. For non-cyclic extensions the local-global principle can fail. This theorem is the archetype of the local-to-global philosophy that pervades modern arithmetic geometry, and it knits the local reciprocity laws into the global theory.

Theorem (conductor-discriminant formula). For a finite abelian extension $L/K$ with Galois group $G$, the relative discriminant ${\mathfrak{d}}_{L/K}$ equals the product of the conductors $\mathfrak{f}(\chi )$ over all characters $\chi \in \hat{G}$: ${\mathfrak{d}}_{L/K}={\prod }_{\chi }\mathfrak{f}(\chi )$. Equivalently the Dedekind zeta function of $L$ factors as ${\zeta }_L(s)={\prod }_{\chi }L(s,\chi )$ over the Hecke $L$-functions attached to the characters of $G$ ^{[Neukirch Ch. VII]}. This links the conductor — the modulus measuring ramification in class field theory — to the analytic factorisation of zeta functions developed in 21.03.03, and is the seed of the non-abelian factorisation predicted by the Langlands programme.

Synthesis. Class field theory is governed by the dictionary between congruence subgroups inside $K$ and abelian extensions above $K$, and the foundational reason the dictionary closes is exactly the two inequalities: the first bounds the index of the norm subgroup from below and the second from above, pinning it to $[L:K]=|G|$. The central insight is that the Artin map — reading the whole Galois group off Frobenius elements at unramified primes — is dual to the local norm-residue pairing of 21.02.05, and this is exactly what forces congruence data inside $K$ to determine splitting data in $L$. The bridge is that the abelian theory is the degree-one layer of the Langlands programme 21.10.01, where abelian characters of the Galois group generalise to automorphic representations; putting these together, the existence theorem shows reciprocity is not merely a property of given extensions but a complete classification, and the whole theory builds toward the non-abelian Galois-representation theory of 21.05.01 where compatible $\ell$-adic families appear again in place of the finite abelian groups classified here.

Full proof set Master

Proposition 1 (the Artin map for cyclotomic fields). Let $L=\mathbb{Q}({\zeta }_n)$, $K=\mathbb{Q}$, and let $(p)$ be a rational prime with $p\nmid n$. The Artin map ${\Phi }_{L/\mathbb{Q}}:I_{\mathbb{Q}}^{(n)}\to \mathrm{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})$ sends $(p)$ to the automorphism ${\zeta }_n\mapsto {\zeta }_n^p$, identifying $\mathrm{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})$ with $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ with kernel the principal ray $P_{\mathbb{Q},1}^{(n)}=\{(a):a\equiv 1(\mathrm{mod}n),a>0\}$.

Proof. Since $p\nmid n$, the prime $(p)$ is unramified in $\mathbb{Q}({\zeta }_n)$: cyclotomic extensions ramify only at primes dividing $n$. Let $\mathfrak{P}$ be a prime of $\mathbb{Z}[{\zeta }_n]$ above $p$, with residue field $\mathbb{Z}[{\zeta }_n]/\mathfrak{P}$. The Artin symbol $\sigma =\left(\frac{L/\mathbb{Q}}{(p)}\right)$ is the unique automorphism of $L$ whose reduction modulo $\mathfrak{P}$ is the Frobenius $\bar{\alpha }\mapsto {\bar{\alpha }}^p$ of $\mathbb{Z}[{\zeta }_n]/\mathfrak{P}$ over ${\mathbb{F}}_p$ (here $N(p)=p$). Applied to $\alpha ={\zeta }_n$,

$$\overline{\sigma ({\zeta }_n)}={\bar{\zeta }}_n^p.$$

The assignment ${\zeta }_n\mapsto {\zeta }_n^p$ is itself a Galois automorphism of $\mathbb{Q}({\zeta }_n)$, because $\gcd (p,n)=1$ and so ${\zeta }_n^p$ is again a primitive $n$th root of unity; call this automorphism ${\tau }_p$. Both $\sigma$ and ${\tau }_p$ reduce modulo $\mathfrak{P}$ to the same Frobenius map $\bar{\alpha }\mapsto {\bar{\alpha }}^p$, and reduction $\mathrm{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})\hookrightarrow \mathrm{Gal}((\mathbb{Z}[{\zeta }_n]/\mathfrak{P})/{\mathbb{F}}_p)$ is injective at unramified primes. Hence $\sigma ={\tau }_p$, that is, ${\Phi }_{L/\mathbb{Q}}((p)):{\zeta }_n\mapsto {\zeta }_n^p$.

The standard identification $\mathrm{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})\stackrel{\sim }{\to }(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ sends $\rho \mapsto [a]$ where $\rho ({\zeta }_n)={\zeta }_n^a$; it is an isomorphism because the cyclotomic polynomial ${\Phi }_n(x)$ is irreducible over $\mathbb{Q}$ (each primitive $n$th root is a conjugate). Under this identification, ${\Phi }_{L/\mathbb{Q}}((p))=[p\mathrm{mod}n]$. For surjectivity, take any class $[a]\in (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ represented by a positive integer $a$ coprime to $n$; the principal ideal $(a)={\prod }_p{p}^{v_p(a)}$ lies in $I_{\mathbb{Q}}^{(n)}$ and ${\Phi }_{L/\mathbb{Q}}((a))={\prod }_p[p{]}^{v_p(a)}=[a]$, so every class is hit. For the kernel, $(a)\in \ker \Phi$ exactly when ${\zeta }_n^a={\zeta }_n$, i.e. $a\equiv 1(\mathrm{mod}n)$, which is the principal ray $P_{\mathbb{Q},1}^{(n)}$. This is the reciprocity isomorphism $I_{\mathbb{Q}}^{(n)}/P_{\mathbb{Q},1}^{(n)}\cong (\mathbb{Z}/n\mathbb{Z}{)}^{\times }\cong \mathrm{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})$ in the founding case $K=\mathbb{Q}$. $\square$

Proposition 2 (the Hilbert class field of $\mathbb{Q}(\sqrt{-5})$ is $\mathbb{Q}(\sqrt{-5},i)$). Let $K=\mathbb{Q}(\sqrt{-5})$, which has class number $h_K=2$ and discriminant $d_K=-20$. Then $H=\mathbb{Q}(\sqrt{-5},i)$ is an abelian extension of $K$ of degree $2$, unramified at every finite prime, and is therefore the Hilbert class field of $K$.

Proof. The extension is abelian because $i\mapsto -i$ generates a group of order $2$ that commutes with the (real) symmetries of $K$. The element $i$ does not lie in $K$ (the imaginary quadratic field $K$ contains only the quadratic subfield $\mathbb{Q}(\sqrt{-5})$ over $\mathbb{Q}$, and $\mathbb{Q}(i)\ne \mathbb{Q}(\sqrt{-5})$), so $[H:K]=2$.

To verify unramification, compute the discriminant of $H$ over $\mathbb{Q}$. The field $H=\mathbb{Q}(\sqrt{-5},i)$ is biquadratic, with three quadratic subfields: $\mathbb{Q}(\sqrt{-5})$, $\mathbb{Q}(i)$, and $\mathbb{Q}(\sqrt{5})$ (the last because $\sqrt{-5}\cdot i=\sqrt{5}$ up to sign). Their discriminants are $-20$ (since $-5\equiv 3(\mathrm{mod}4)$), $-4$, and $5$ (since $5\equiv 1(\mathrm{mod}4)$). The discriminant of a biquadratic field equals the product of the discriminants of its three quadratic subfields, so

$$d_H=(-20)\cdot (-4)\cdot 5=400=(-20{)}^2=d_K^2.$$

The relation between absolute, base, and relative discriminants reads $d_H=d_K^{[H:K]}{N}_{K/\mathbb{Q}}({\mathfrak{d}}_{H/K})$, i.e. $d_H=d_K^{2}N({\mathfrak{d}}_{H/K})$. Substituting $d_H=d_K^2$ gives $N({\mathfrak{d}}_{H/K})=1$, hence the relative discriminant ideal ${\mathfrak{d}}_{H/K}={\mathcal{O}}_K$. A prime of $K$ ramifies in $H$ exactly when it divides ${\mathfrak{d}}_{H/K}$, so no finite prime ramifies. The field $K$ is totally imaginary (it has no real embeddings), so there are no infinite places to consider. Thus $H/K$ is unramified at every prime.

Finally, $H/K$ is an unramified abelian extension of degree $2=h_K$. By Artin reciprocity the maximal unramified abelian extension of $K$ has degree $h_K=2$, so $H$ realises that maximum and $H$ is the Hilbert class field. $\square$

Proposition 3 (local reciprocity for unramified extensions of $p$-adic fields). Let $L/K$ be a finite unramified extension of $p$-adic fields of degree $n$. Then $N_{L/K}(L^{\times })=⟨{\pi }_K^n⟩\cdot {\mathcal{O}}_K^{\times }$, and the local Artin map induces $K^{\times }/N_{L/K}(L^{\times })\cong \mathrm{Gal}(L/K)$ sending the class of a uniformiser ${\pi }_K$ to the Frobenius generator.

Proof. Because $L/K$ is unramified, a uniformiser ${\pi }_K$ of $K$ is also a uniformiser of $L$, the ramification index is $e=1$, and the residue-field extension is ${\mathbb{F}}_{q^n}/{\mathbb{F}}_q$ of degree $f=n$ where $q=|{\mathcal{O}}_K/{\mathfrak{m}}_K|$. The Galois group $\mathrm{Gal}(L/K)$ is canonically $\mathrm{Gal}({\mathbb{F}}_{q^n}/{\mathbb{F}}_q)$, cyclic of order $n$, generated by the Frobenius $x\mapsto x^q$.

Norms of uniformisers. Since ${\pi }_K\in L$ is a uniformiser with $v_L({\pi }_K)=1$ and $[L:K]=n$, the valuation formula gives $v_K(N_{L/K}({\pi }_K))=f\cdot v_L({\pi }_K)=n\cdot 1=n$, so $N_{L/K}({\pi }_K)=u{\pi }_K^n$ for a unit $u\in {\mathcal{O}}_K^{\times }$. Hence the uniformiser-part of the norm group is $⟨{\pi }_K^n⟩$ (up to units).

Norms of units. The residue-level norm $N:{\mathbb{F}}_{q^n}^{\times }\to {\mathbb{F}}_q^{\times }$ is $\bar{v}\mapsto {\bar{v}}^{1+q+\cdots +q^{n-1}}$, a surjection between cyclic groups (its image is all of ${\mathbb{F}}_q^{\times }$ by finite-field theory). Given $u\in {\mathcal{O}}_K^{\times }$ with residue $\bar{u}\in {\mathbb{F}}_q^{\times }$, pick $\bar{v}\in {\mathbb{F}}_{q^n}^{\times }$ with $N(\bar{v})=\bar{u}$, lift $\bar{v}$ to $v\in {\mathcal{O}}_L^{\times }$, and apply Hensel's lemma to the polynomial equation $N_{L/K}(X)-u=0$ (which has the approximate root $v$ mod ${\mathfrak{m}}_L$) to obtain a true lift $w\in {\mathcal{O}}_L^{\times }$ with $N_{L/K}(w)=u$. Hence ${\mathcal{O}}_K^{\times }\subseteq N_{L/K}({\mathcal{O}}_L^{\times })$.

Combining the two parts, $N_{L/K}(L^{\times })=⟨{\pi }_K^n⟩\cdot {\mathcal{O}}_K^{\times }$. The quotient

$$K^{\times }/N_{L/K}(L^{\times })\cong ⟨{\pi }_K⟩\cdot {\mathcal{O}}_K^{\times }/⟨{\pi }_K^n⟩\cdot {\mathcal{O}}_K^{\times }\cong ⟨{\pi }_K⟩/⟨{\pi }_K^n⟩\cong \mathbb{Z}/n\mathbb{Z}$$

is cyclic of order $n$, generated by the class of ${\pi }_K$. Sending this class to the Frobenius generator of $\mathrm{Gal}(L/K)\cong \mathbb{Z}/n\mathbb{Z}$ gives the local reciprocity isomorphism. $\square$

Connections Master

• Algebraic number theory 21.17.01. Class field theory is built directly on the ideal-theoretic foundations of 21.17.01: the ray class group ${\mathrm{Cl}}_{\mathfrak{m}}(K)$ is a quotient of the ideal group $I_K^{\mathfrak{m}}$ by a refinement of the principal ideals $P_K$, reducing to the ordinary class group $\mathrm{Cl}(K)$ when $\mathfrak{m}=1$. The Hilbert class field is the extension-theoretic avatar of that class group, and the discriminant formula $d_H=d_K^{[H:K]}N({\mathfrak{d}}_{H/K})$ used throughout this unit is the relative form of the ramification theory developed there.

• Fields and Galois theory 01.04.01. The Galois group $\mathrm{Gal}(L/K)$ that carries the Artin map is the central object of 01.04.01, and the whole theory is restricted to its abelian instances — finite Galois extensions whose group is abelian. Frobenius elements, decomposition groups, and the splitting behaviour of primes underlie every statement here and are imported wholesale from the field-theoretic foundations.

• Galois representations 21.05.01. The Artin map is the one-dimensional, finite-image case of a Galois representation: a continuous homomorphism $\mathrm{Gal}(\bar{K}/K)\to {\mathbb{C}}^{\times }$, factoring through a finite abelian quotient. The compatible families of $\ell$-adic Galois representations of 21.05.01 are the higher-dimensional, non-finite-image descendants of these abelian characters, and class field theory classifies precisely the degree-one layer they generalise.

• The Langlands programme 21.10.01. Replacing the abelian Galois group by the full non-abelian group and finite abelian characters by automorphic representations of ${\mathrm{GL}}_n$ produces the Langlands correspondence of 21.10.01. Class field theory is thus the $n=1$ case of Langlands, and the reciprocity map $I_K^{\mathfrak{m}}/P_{K,1}^{\mathfrak{m}}N(I_L^{\mathfrak{m}})\cong \mathrm{Gal}(L/K)$ is the abelian shadow of the non-abelian functoriality predicted there.

• Local fields and the Hilbert symbol 21.02.05. The global reciprocity law is assembled from local reciprocity laws at each place of $K$, and the local norm-residue computation underlying Stage 2 of the reciprocity proof runs on the Hilbert symbol of 21.02.05. The Hasse norm theorem likewise rests on the local-global principles for quadratic forms established in the local-field theory of 21.02.03 and 21.02.05.

• Analytic number theory: the class-number formula 21.03.03. The conductor-discriminant formula factors ${\zeta }_L(s)$ over Hecke $L$-functions of the characters of $\mathrm{Gal}(L/K)$, the analytic theory developed in 21.03.03, and the class number $h_K$ that measures the Hilbert class field is recovered from the residue of ${\zeta }_K$ at $s=1$. This unit supplies the class-field-theoretic structure that the analytic class-number formula of 21.03.03 evaluates.

Historical & philosophical context Master

Class field theory was completed over four decades by a single sustained programme. Hilbert's Zahlbericht of 1897 ^{[Hilbert 1897]} set the agenda by listing the problems — the existence of the Hilbert class field, the factorisation of primes under extension, the reciprocity laws — whose solution would constitute the theory. Hilbert conjectured, and Furtwängler later proved, the principalisation theorem: every ideal of $K$ becomes principal in the Hilbert class field $H_K$.

The decisive breakthrough was Takagi's 1920 paper Über eine Theorie des relativ-Abel'schen Zahlkörpers ^{[Takagi 1920]}, which proved the existence and uniqueness theorems for ray class fields and established the bijection between congruence subgroups and abelian extensions. Takagi's theory described which abelian extensions existed but not how the Galois group matched the ray class group — the isomorphism was known to exist abstractly but had no canonical description.

That final step was Artin's. In the 1927 paper Beweis des allgemeinen Reziprozitätsgesetzes ^{[Artin 1927]}, Artin introduced the Frobenius symbol — now the Artin symbol — and proved it realises the reciprocity isomorphism explicitly, completing the theory. Hasse's Bericht of 1930 ^{[Hasse 1930]} systematised the local-global picture, including the norm theorem for cyclic extensions that bears his name. Chevalley's 1940 monograph La théorie du corps de classes ^{[Chevalley 1940]} then introduced the idèle group, replacing the ideal-theoretic ray class groups with idele class groups and recasting the entire theory in cohomological form, which removed the dependence on analytic estimates and exposed the first and second inequalities as the structural engine.

The conceptual move of class field theory — that the symmetries of a field are classified by arithmetic data inside it — became the template for the Langlands programme, which seeks the same kind of dictionary for non-abelian extensions. The abelian theory remains the one corner of that larger programme that is fully closed, and its completion stands as the crown jewel of algebraic number theory.

Bibliography Master

@article{Hilbert1897,
  author  = {Hilbert, David},
  title   = {Die Theorie der algebraischen Zahlk{\"o}rper},
  journal = {Jahresbericht der Deutschen Mathematiker-Vereinigung},
  volume  = {4},
  pages   = {175--546},
  year    = {1897}
}

@article{Takagi1920,
  author  = {Takagi, Teiji},
  title   = {{\"U}ber eine Theorie des relativ-{A}bel'schen Zahlk{\"o}rpers},
  journal = {Journal of the College of Science, Imperial University of Tokyo},
  volume  = {41},
  number  = {9},
  pages   = {1--133},
  year    = {1920}
}

@article{Artin1927,
  author  = {Artin, Emil},
  title   = {Beweis des allgemeinen Reziprozit{\"a}tsgesetzes},
  journal = {Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg},
  volume  = {5},
  pages   = {353--363},
  year    = {1927}
}

@book{Hasse1930,
  author    = {Hasse, Helmut},
  title     = {Bericht {\"u}ber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlk{\"o}rper},
  publisher = {Teubner},
  address   = {Leipzig},
  series    = {Jahresbericht der DMV, Erg{\"a}nzungsband},
  volume    = {6},
  year      = {1930}
}

@book{Chevalley1940,
  author    = {Chevalley, Claude},
  title     = {La th{\'e}orie du corps de classes},
  publisher = {Princeton University Press},
  series    = {Annals of Mathematics Studies},
  number    = {3},
  year      = {1940}
}

@book{Neukirch1999,
  author    = {Neukirch, J{\"u}rgen},
  title     = {Algebraic Number Theory},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {322},
  year      = {1999},
  note      = {Translated by Norbert Schappacher}
}

@book{Serre1979,
  author    = {Serre, Jean-Pierre},
  title     = {Local Fields},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {67},
  year      = {1979},
  note      = {Translated by Marvin J. Greenberg}
}

@book{ArtinTate2009,
  author    = {Artin, Emil and Tate, John},
  title     = {Class Field Theory},
  publisher = {American Mathematical Society},
  series    = {AMS Chelsea Publishing},
  year      = {2009}
}

@book{Childress2009,
  author    = {Childress, Nancy},
  title     = {Class Field Theory},
  publisher = {Springer-Verlag},
  series    = {Universitext},
  year      = {2009}
}

@book{Cox1989,
  author    = {Cox, David A.},
  title     = {Primes of the Form $x^2 + n y^2$: Fermat, Class Field Theory, and Complex Multiplication},
  publisher = {Wiley-Interscience},
  year      = {1989}
}