The principal genus theorem

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Public domain (CC0).

For a quadratic field, genus characters record congruence information about ideal classes. The principal genus theorem says exactly how much information they lose: its kernel consists of squares. For a general Galois extension, Noether replaces one ideal class by a family of classes and its multiplication defects. The arithmetic condition on those defects is local splitting of a crossed-product algebra. We prove her reduction with that condition intact.

Take first Hilbert 90 in Noether's form and Galois descent, Discrete valuation rings and Dedekind domains, and Hilbert's ramification theory in Galois extensions. Recall the particular facts needed from the latter: the Galois group acts transitively on primes above a prime of the ground field; the stabilizer is its decomposition group; the completed extension has this group; and an unramified completed extension is cyclic, generated by residue-field Frobenius. We also use completions and ideal norms. The elementary ideal-class and norm background is available in Norms of ideals, the ideal class group, and modules over Dedekind domains.

Throughout, \(L/K\) is a finite Galois extension of number fields, \(G=\operatorname{Gal}(L/K)\), and group actions are on the left. Let \(I_L\) be the multiplicative group of nonzero fractional ideals, \(P_L\) its principal-ideal subgroup, and \(\operatorname{Cl}(L)=I_L/P_L\) its ordinary class group. At real places, positivity will enter local norm conditions explicitly. We do not silently replace an ordinary class group by a narrow one. In an imaginary quadratic field they coincide.

1. Arithmetic theorems and their proof order

The algebraic prerequisites are Central simple algebras and the Brauer group and Crossed products and factor systems. The arithmetic results below are proved in the class-field-theory lessons named here, after those algebraic prerequisites and before their use in this lesson. Their proofs do not use the principal-genus argument developed below.

Hasse norm theorem. Let \(L/K\) be a finite cyclic extension of number fields. An element \(a\in K^\times\) is a field norm if and only if it is a norm from \(L_w/K_v\) at every place \(v\), for one, equivalently every, \(w\mid v\). The proof is The norm index bound and Hasse's norm theorem, Theorem 15.3: the cyclic idèle-class norm-index calculation makes the field-norm quotient inject into the idèle-norm quotient. A family of local norm preimages is an idèle after choosing units outside finitely many places. Norms from the factors of \(L\otimes_KK_v\) have the same image, so the formulation using one completion agrees with the product formulation.

Brauer–Hasse–Noether splitting theorem. For every central simple \(K\)-algebra \(A\),

The general injectivity proof is Brauer groups of local and global fields, section 4. It first treats cyclic relative classes by the norm theorem, then a normal subgroup of index \(p\) in a \(p\)-group, and finally Sylow restriction and corestriction. Thus it does not assume that a global division algebra has a cyclic maximal subfield. Its restriction–corestriction identities are proved in section 1 of that lesson. Splitting over \(K\) plainly implies splitting after completion, which supplies the other implication of (1).

For the quadratic examples we use Quadratic fields: ideal classes and binary quadratic forms: Theorem 10.1 identifies proper form classes with oriented ideal classes, and Theorem 10.2 proves positive-definite reduction with its boundary signs. Its Theorem 10.6 identifies the genus characters, including the dyadic cases. Only the description of these characters is needed here; the independent two-torsion argument below proves their kernel. In the real quadratic case that provider uses narrow classes; our quadratic fields are imaginary, where narrow and ordinary classes agree. The provider's class-field input is the narrow Hilbert class field of Hilbert and ring class fields, Theorem 19.1, section 1, obtained from ray-class existence, rather than the later genus calculation that uses quadratic forms.

Quadratic reciprocity and both supplementary laws are proved in Cyclotomic fields, Theorem 12.5. Dirichlet's theorem on primes in an invertible residue class is proved in Dirichlet's theorem on primes in arithmetic progressions. These statements supply respectively the prime-discriminant character conventions and the independence of those characters. Reduction gives finiteness of the imaginary quadratic class group, since \(3A^2\le |D|\) leaves only finitely many reduced forms. None of this assumes the kernel assertion proved next.

2. The imaginary quadratic principal genus

Let \(L=\mathbb Q(\sqrt d)\), with \(d<0\) squarefree, and let \(D\) be its fundamental discriminant. Factor

into distinct prime discriminants: for an odd ramified prime \(p\), the factor is \(p^*=(-1)^{(p-1)/2}p\); there is at most one even factor, in \(\{-4,8,-8\}\). Thus \(t\) is the number of ramified rational primes. Write \(\chi_{d_i}(m)=(d_i/m)\) for the quadratic Kronecker character, evaluated on positive integers coprime to \(D\).

An ideal class has an integral representative \(\mathfrak a\) coprime to the ramified primes. The genus characters are

We use the standard characterization of quadratic genera by these characters: two ideal classes, or their corresponding primitive positive definite forms, are in the same genus exactly when their character values agree. This is Gauss's genus-character description. Its identity fiber is the principal genus.

Here are the elementary features of (2) used in the proof. Multiplicativity of ideal norms makes the characters homomorphisms. Principal ideals contribute value \(1\): at an odd ramified \(p\), a field norm of a local unit reduces to a square modulo \(p\), so \(\chi_{p^*}(N\alpha)=(N\alpha/p)=1\). At \(2\), the same check can be made directly. If \(d\) is odd with even discriminant, \(N(a+b\sqrt d)=a^2-db^2\), when odd, is \(1\bmod4\). If \(d=2m\), an odd norm is \(1\) or \(1-2m\bmod8\): for \(m\equiv1\bmod4\) these are \(1,7\), on which \(\chi_8=1\); for \(m\equiv3\bmod4\) they are \(1,3\), on which \(\chi_{-8}=1\). These checks apply to local units and hence to fractional principal quotients as well. For odd discriminant, writing \(\alpha=(a+b\sqrt d)/2\) gives the same square calculation at every odd ramified prime. Thus the values are independent of the ideal representative. Representatives avoiding finitely many primes exist by ideal approximation, or the Chinese remainder theorem in a Dedekind domain.

The product of all characters is \(1\). Indeed, factor a coprime ideal into unramified prime ideals. A prime above a split rational prime \(\ell\) has norm \(\ell\), with \((D/\ell)=1\); a prime above an inert \(\ell\) has norm \(\ell^2\), also of character \(1\). Therefore (2) defines

Theorem 2.1. The map (3) is surjective, and

Thus there are \(2^{t-1}\) genera, and the imaginary quadratic principal genus consists exactly of squares.

Proof. First prove surjectivity. At each odd \(p\mid D\), the character \((m/p)\) takes both signs on invertible residues. The even character, when present, takes both signs on invertible residues modulo \(4\) or \(8\). Their moduli are pairwise coprime, so the Chinese remainder theorem prescribes their signs independently on a residue \(m\) coprime to \(D\). For any desired tuple in \(E\), the resulting residue has \((D/m)=1\). Dirichlet's theorem gives a positive prime \(\ell\) in that residue class. It splits in \(L\), and either prime ideal of norm \(\ell\) has exactly the desired tuple. This proves surjectivity. For \(t=1\), the target is already the trivial group.

Every square lies in the kernel because every character has order at most two. To prove equality, we bound the subgroup \(\operatorname{Cl}(L)[2]\) of elements of order dividing two.

Conjugation acts on the class group by inversion: \(\mathfrak a\overline{\mathfrak a}=(N\mathfrak a)\mathcal O_L\). Thus a class of order at most two is fixed by conjugation. Choose its representative \(\mathfrak a\); then \(\overline{\mathfrak a}=(\alpha)\mathfrak a\) for some \(\alpha\in L^\times\). Taking absolute ideal norms gives \(N_{L/\mathbb Q}(\alpha)=1\), since this norm is positive. Scalar Hilbert 90 gives \(\alpha=x/\overline x\). Consequently \(x\mathfrak a\) is fixed by conjugation.

In a conjugation-invariant fractional ideal, the two primes above a split prime occur with equal exponents and give a rational principal ideal. An inert prime itself is a rational principal ideal. Only ramified prime ideals can contribute to its class. Thus \(\operatorname{Cl}(L)[2]\) is generated by the \(t\) ramified prime classes \([\mathfrak p_i]\), each of exponent at most two since \(\mathfrak p_i^2=(p_i)\).

There is at least one nonempty relation among these generators. If \(d\ne-1\), then

because the valuation is one at each prime dividing the squarefree radicand and zero elsewhere. This product is nonempty. If \(d=-1\), the sole ramified prime is already principal, generated by \(1+i\). Therefore

For any finite abelian group \(C\), the squaring map has kernel \(C[2]\) and image \(C^2\), so \(|C/C^2|=|C[2]|\). Apply this to \(C=\operatorname{Cl}(L)\). Its quotient by squares maps surjectively to \(E\), which has \(2^{t-1}\) elements, and has at most that many elements. The map is therefore bijective. Its kernel before quotienting is exactly \(C^2\). \(\square\)

Notice where the converse was proved: generation of the two-torsion by ramified primes supplies the upper bound, and independent genus characters supply the lower bound. Merely observing that characters kill squares would prove only one inclusion.

Two complete genus calculations

A reduced primitive positive definite form \((A,B,C)\) has discriminant \(B^2-4AC=D\), with \(|B|\le A\le C\), and \(B\ge0\) if \(|B|=A\) or \(A=C\). In particular \(3A^2\le|D|\). The reduction theorem of Lagrange and Gauss gives exactly one such form per proper class, with these boundary conventions.

For \(D=-20=(-4)\cdot5\), the bound gives \(A\le2\). Testing \(A=1,2\), even \(B\), and \(C=(B^2+20)/(4A)\) gives exactly \((1,0,5)\), \((2,2,3)\). Their character values can be read from the coprime numbers they represent:

The two distinct classes give \(\operatorname{Cl}(\mathbb Q(\sqrt{-5}))\simeq\mathbb Z/2\). Each genus has one class, and the principal genus is the identity, equal to the squares.

For \(D=-84=(-4)(-3)(-7)\), the bound gives \(A\le5\). The same finite check gives exactly \((1,0,21)\), \((2,2,11)\), \((3,0,7)\), \((5,4,5)\). For clarity, \(A=4\) produces no integral \(C\) with the required parity and bounds, and at \(A=5\) the solutions \(B=\pm4,C=5\) are one boundary class with \(B=4\).

Here \(19=3\cdot2^2+7\). For the odd characters, \(\chi_{-3}(m)=(m/3)\), \(\chi_{-7}(m)=(m/7)\); the residues \(11\equiv4\), \(19\equiv5\), \(5\equiv5\pmod7\) give the displayed signs. All four possible tuples occur on the four classes. Thus the character map is an isomorphism onto \((\mathbb Z/2)^2\); the class group is that group, there are four genera, and the principal genus again has one class. This determines the group law without guessing from the class number alone.

3. Ideal cocycles and the cyclic theorem

For a family \(\mathfrak c_\sigma\in I_L\), write

The identity \(\delta\mathfrak c=1\) is the ideal-valued 1-cocycle condition.

Lemma 3.1 (Artin's ideal argument). Every ideal-valued 1-cocycle has the form

Proof. Define \(\mathfrak b=\sum_{\tau\in G}\mathfrak c_\tau\), the sum of the finitely many fractional ideals as \(\mathcal O_L\)-submodules. It is a nonzero fractional ideal. Multiplication by an invertible ideal distributes over sums, so (4) gives

This is (5). For prime exponents, the sum takes the minimum of the finitely many exponents; it is the ideal-theoretic greatest common divisor. The same argument is Noether's first lemma in work 41 §2.3. \(\square\)

Now suppose \(G=\langle\sigma\rangle\) is cyclic of order \(n\). For a place \(v\) ramified in \(L/K\), including a real place becoming complex, call \(\alpha\in K^\times\) a norm residue at \(v\) if \(\alpha\in N_{L_w/K_v}(L_w^\times)\).

Theorem 3.2 (cyclic principal genus, with its norm-residue condition). Suppose

and at least one such generator \(\alpha\) is a norm residue at every ramified place. Then

for some \(\beta\in L^\times\), \(\mathfrak b\in I_L\). Conversely (6) supplies a generator \(N(\beta)\) with those norm-residue properties.

Proof. At an unramified finite \(v\), put \(f=[L_w:K_v]\). Prime ideal factorization gives

so \(v(\alpha)\) is divisible by \(f\). In an unramified local extension every unit is a norm, as proved in Lemma 4.1 below, and a ground-field uniformizer has norm its \(f\)-th power. Hence \(\alpha\) is a local norm at this place too. At an unramified infinite place the local extension is trivial; the only nontrivial infinite case is already among the imposed real-to-complex conditions. Thus \(\alpha\) is a local norm everywhere. The cyclic Hasse norm theorem of section 1 gives \(\beta\in L^\times\) with \(N(\beta)=\alpha\).

Put \(\mathfrak d=\mathfrak a/(\beta)\); then its ideal norm is \(1\), and

This identity follows prime by prime from extension of the ideal norm: the product has exponent \(ef\sum_{w\mid v}w(\mathfrak d)\) at every \(w\mid v\). Define \(\mathfrak c_{\sigma^i}=\prod_{j=0}^{i-1}\sigma^j(\mathfrak d)\), \(0\le i<n\). The norm-one identity handles a carry of exponents, so these ideals satisfy (4) with value \(1\). Lemma 3.1 gives \(\mathfrak d=\mathfrak b/\sigma(\mathfrak b)\), proving (6). Conversely its ideal norm is \((N\beta)\), and a global field norm is a local norm at every place. \(\square\)

The generator qualification matters: replacing \(\alpha\) by an arbitrary unit multiple need not preserve each norm-residue condition. On classes, the principal genus defined by this condition is exactly \((1-\sigma)\operatorname{Cl}(L)\). Indeed the condition is unchanged when \(\mathfrak a\) is multiplied by a principal ideal, because its norm generator is multiplied by a global norm. Theorem 3.2 proves both inclusions. For an imaginary quadratic extension of \(\mathbb Q\), conjugation acts by inversion, so \((1-\sigma)\operatorname{Cl}(L)=\operatorname{Cl}(L)^2\), agreeing with section 2.

Congruences and the cyclic ray principal genus

A ray class group \(\operatorname{Cl}_{\mathfrak M}(L)\) uses fractional ideals prime to its finite modulus, modulo principal ideals with a generator congruent to \(1\) at that modulus and positive at any designated real places. We use a finite integral ground modulus \(\mathfrak m\). Let \(\mathfrak f\) be the conductor of \(L/K\), with each real place becoming complex in its infinite part. Define the finite genus modulus by

Here \(\mathfrak D_{L/K}\) is the different. The upper modulus in (R1) is finite; the required real-place signs occur in the ground conductor. We first prove the local norm assertion explaining this normalization.

Lemma 3.3 (norms beyond the conductor). Let \(E/F\) be a finite cyclic extension of nonarchimedean local fields, with ramification index \(e\), different exponent \(d\) in \(E\), and conductor exponent \(c\) in \(F\). Put \(r_0=ec-d\). For every integer \(m\ge0\),

Proof. For an unramified extension, \(c=d=0\) and \(e=1\). Residue norms are onto. The successive trace corrections in Lemma 4.1, starting at any positive level \(m\), prove (R2), including \(m=0\).

Suppose the extension is ramified. Use the upper and lower numbering of Herbrand's function and the upper numbering, Theorem 3.1 and the tower formula, and the conductor and Hasse–Arf theorem of Abelian ramification, conductors and Hasse–Arf, Proposition 10.4 and Corollary 10.3. If \(t\) is the last lower break, those results give

For the middle identity, integrate the ramification filtration: \(e\varphi(t)=\int_0^t|G_x|\,dx\), while the different formula gives \(d=(e-1)+\int_0^t(|G_x|-1)\,dx\). Thus \(d=e(\varphi(t)+1)-t-1\). In particular the genus exponent is positive at every ramified prime.

We prove the equality of compact unit subgroups in (R2) by testing finite characters. Norms of \(U_E^r\) are open. The nondegenerate trace pairing gives \(z\in E\) of trace one, so write \(N(1+Tz)=1+T+\sum_{i\ge2}b_iT^i\). Choose an integer \(h\ge1\) such that \(eh+v_E(z)\ge r\) and \(v_F(b_i)+(i-1)h\ge1\) for every nonzero \(b_i\). For \(a\in\mathfrak p_F^h\), the substitution \(T\mapsto a-\sum_{i\ge2}b_iT^i\) preserves \(\mathfrak p_F^h\) and increases the valuation of the difference of two arguments by at least one: factor each \(T^i-S^i\) by \(T-S\). Iteration is Cauchy and converges by completeness to a solution of \(N(1+Tz)=1+a\). The first inequality puts \(1+Tz\) in \(U_E^r\), so the norm image contains a principal-unit neighborhood. The image is compact by compactness of \(U_E^r\), hence closed.

Let \(\chi\) be any continuous finite-order character of \(F^\times\). Local reciprocity and its norm compatibility are proved in Explicit local reciprocity and the existence theorem. Let \(T/F\) be the abelian extension through which \(\chi\) factors, put \(B=\operatorname{Gal}(TE/F)\), and \(H=\operatorname{Gal}(TE/E)\). Set \(s=c+m\), \(r=r_0+em\), and

Above the last break \(\psi_{E/F}\) is affine of slope \(e\). Equations (R3) give \(\psi_{E/F}(v)=r\). The subgroup and tower formulas give

Since \(v>c-1\), the image of \(B^v\) in \(\operatorname{Gal}(E/F)\) is trivial; hence \(B^v\subset H\). As \(TE/F\) is abelian, all its upper jumps are integers. Since \(s-1<v\le s\), one has \(B^v=B^s\). Unit reciprocity identifies the images of \(U_E^r\) and \(U_F^s\) with \(H^r\) and \(B^s\), respectively. Norm compatibility consequently proves

These character tests determine the subgroups. Indeed both lie in \(U_F\) and contain an open principal-unit subgroup. The quotient by a common such subgroup is finite abelian, and its characters separate every subgroup. One can extend a character of this finite unit quotient to \(F^\times=\pi^{\mathbb Z}\times U_F\) by assigning value one to \(\pi\). Characters separate a finite abelian quotient because decomposition into cyclic groups supplies a character nontrivial on any prescribed nonzero element. This proves (R2). \(\square\)

Theorem 3.4 (cyclic ray principal genus). Let \(L/K\) be cyclic with generator \(\sigma\), and choose fractional ideals \(\mathfrak a\) prime to \(\mathfrak M\). Then

with \(\mathfrak b\) also prime to \(\mathfrak M\). On the left, congruence includes positivity at each real place becoming complex. Thus the kernel of the norm map from \(\operatorname{Cl}_{\mathfrak M}(L)\) to \(\operatorname{Cl}_{\mathfrak m\mathfrak f}(K)\) is \((1-\sigma)\operatorname{Cl}_{\mathfrak M}(L)\).

Proof. At a finite modulus prime \(v\), the exponent of \(\mathfrak M\) at \(w\mid v\) is \(e(m_v+c_v)-d_w\). Lemma 3.3 gives a local preimage of \(\alpha\) congruent to \(1\) at precisely this level. At a ramified real place, positivity gives a complex norm. At every other place the ideal-norm valuation and Lemma 4.1 make \(\alpha\) a local norm as in Theorem 3.2. Hasse's theorem therefore supplies \(\beta_0\in L^\times\) with \(N\beta_0=\alpha\).

We may arrange the required local congruences without changing this norm. At each finite modulus place choose a vector \(z_v\in(L\otimes_KK_v)^\times\) of product norm \(\alpha\), congruent to one in all its factors: choose the local preimage just constructed in one factor and one in the others. Then \(z_v/\beta_0\) has product norm one. Hilbert 90 for this cyclic product algebra writes it as \(b_v/\sigma(b_v)\). To see this product version explicitly, list the factors around the \(\sigma\)-orbit. Choose the first coordinate of \(b_v\), determine the succeeding coordinates by the prescribed ratios, and use the norm-one condition after one full orbit to obtain the norm-one equation for the stabilizer on the first coordinate. Scalar Hilbert 90 for that cyclic stabilizer solves this last equation.

Weak approximation chooses \(b\in L^\times\) sufficiently close to every coordinate of these finitely many \(b_v\). Inversion and multiplication are continuous, so \(\beta=\beta_0b/\sigma(b)\) then has \(\beta\equiv1\pmod{\mathfrak M}\) and still \(N\beta=\alpha\). Weak approximation is proved in Absolute values, valuations and Ostrowski's theorem, Theorem 5.2. Its statement with field-valued targets extends to targets in the completions by density in the defining completion.

Now \(\mathfrak d=\mathfrak a/(\beta)\) is prime to \(\mathfrak M\) and has ideal norm one. The cyclic cocycle construction in Theorem 3.2 and Artin's sum in Lemma 3.1 give \(\mathfrak d=\mathfrak b/\sigma(\mathfrak b)\). Every ideal in that sum has valuation zero at the modulus primes, so its sum does too; hence \(\mathfrak b\) is prime to \(\mathfrak M\). This proves the forward implication. Conversely, the ideal norm of the right side of (R4) is \((N\beta)\). Formula (R2) puts each local norm in \(U_v^{m_v+c_v}\), and complex norms are positive. Their product therefore gives exactly the ground ray condition. The same calculation makes the norm map on ray classes well-defined and proves the stated kernel. \(\square\)

The modulus relation is the one in Terada's formulation of Hasse's cyclic theorem. Omitting the conductor changes the assertion: for \(L=\mathbb Q(\sqrt{-5})\), trivial upper and lower moduli would make every ideal norm principal in \(\mathbb Q\), whereas \((1-\sigma)\operatorname{Cl}(L)=1\) and \(\operatorname{Cl}(L)\) has two elements. This is a mathematical distinction between the two norm conditions.

A cyclic cubic calculation

Use the field of lesson 5, with \(\theta=\zeta_7+\zeta_7^{-1}\), minimal polynomial \(f(T)=T^3+T^2-2T-1\), and \(\sigma(\theta)=\theta^2-2\). Its group is cyclic of order three. Formula (6) says that every ideal whose norm has a generator satisfying the ramified local norm conditions becomes \(\mathfrak b/\sigma(\mathfrak b)\) after removing one principal factor.

The exact norm identity \(N(t-\theta)=f(t)\) gives

Thus the ideals generated by these two elements have norms \((7)\) and \((13)\), respectively, and their generators are local norms at every place. Their instances of (6) use \(\beta=2-\theta\) or \(\theta+3\), and \(\mathfrak b=\mathcal O_L\). The scalar \(2\), in contrast, is not a local norm at the inert unramified prime \(2\): the residue polynomial \(T^3+T^2+1\) is irreducible over \(\mathbb F_2\), so local norm valuations are multiples of three. This was the obstruction making the parameter-2 cyclic algebra nonsplit in lesson 5. The global norm \(8=N(2)\) passes that valuation test.

4. Noether's theorem for a general Galois extension

We first prove the local algebra statements needed to preserve the precise hypothesis of work 41 §2.

Lemma 4.1 (unramified local norms). For a finite unramified extension \(E/F\) of nonarchimedean local fields, of degree \(f\), the norm on units is surjective. The full norm subgroup consists of elements whose normalized valuation is divisible by \(f\).

Proof. A uniformizer \(\pi\) of \(F\) is also a uniformizer of \(E\), with norm \(\pi^f\). The residue extension is \(\mathbb F_{q^f}/\mathbb F_q\). Its multiplicative norm is surjective by the finite-field calculation in lesson 6, so a target unit can first be matched modulo \(\pi\).

For \(m\ge1\) and \(x\in\mathcal O_E\), expansion of the product of the conjugates gives

The residue trace is surjective: its finite-field extension is separable, so its trace pairing is nondegenerate, and the functional \(x\mapsto\operatorname{Tr}(x)\) is nonzero. Thus, after matching a target unit through \(\pi^m\), multiply the current lift by a suitable \(1+\pi^m x\) to match through \(\pi^{m+1}\). These corrections converge in the complete field \(E\); their product is a unit whose norm, by continuity of the finite product defining it, is the exact target. Finally \(v_F(Ny)=f v_E(y)\), so a norm valuation is divisible by \(f\), and uniformizer powers together with the unit assertion give every such element. \(\square\)

Lemma 4.2 (the local corner of a crossed product). Let \(a\) be an \(L^\times\)-valued normalized 2-cocycle, and \(A=A(a)\). For a place \(v\) and a chosen \(w\mid v\), let \(H=G_w\) and \(E=L_w\), \(F=K_v\). Then

Proof. The coefficient algebra after completion is \(L\otimes_K F=\prod_{w'\mid v}L_{w'}\). Let \(e\) be its primitive idempotent selecting \(w\). Conjugation by \(u_g\) sends the idempotent at \(w\) to the one at \(gw\), so \(e u_g e=0\) unless \(g\in H\). For \(h\in H\), the unit commutes with \(e\). Therefore the corner \(e(A\otimes F)e\) has coefficient field \(E\), basis units \(e u_h\), and exactly the restricted factor system. It is the right-hand crossed product. The completed algebra is central simple, by scalar extension as proved in lesson 4. A nonzero idempotent corner in a matrix algebra over a division ring has the same division representative, hence the same Brauer class, as proved in lesson 5. This proves (7). \(\square\)

Lemma 4.3 (Noether's splitting lemma). Suppose a field-valued factor system \(a\) satisfies

for fractional ideals \(\mathfrak c_\sigma\). If \(A(a)\) splits at every place ramified in \(L/K\), including infinite ramification, then it splits globally. Consequently \(a=\delta d\) for elements \(d_\sigma\in L^\times\).

Proof. Fix an unramified finite \(v\), choose \(w\mid v\), and restrict to \(H=G_w\). In \(E=L_w\), every fractional ideal is principal; choose \(c_h\in E^\times\) with \((c_h)=\mathfrak c_h\mathcal O_E\). Equation (8), restricted to \(H\), says

has valuation zero. It is a unit-valued 2-cocycle, cohomologous to the restricted \(a\). If necessary normalize the cochain at the identity; since \((\mathfrak c_1)=\mathcal O_E\), this only changes units. The group \(H\) is cyclic. In its crossed-product algebra, choose \(u=u_h\) for a generator \(h\), and replace all basis units by \(1,u,\ldots,u^{f-1}\). Their scalar rescalings are products of unit factors \(e\), so \(u^f=t\) is a unit of \(F\). Lemma 4.1 makes \(t\) a norm from \(E\); the cyclic splitting criterion of lesson 5 then makes this local crossed product split. Lemma 4.2 makes \(A(a)\otimes F\) split too.

At an unramified infinite place, the local coefficient extension is trivial, so (7) gives splitting. The remaining places split by hypothesis. The local–global theorem (1) now makes \(A(a)\) split over \(K\). Lesson 5's relative crossed-product classification says precisely that its cocycle is a coboundary, \(a(\sigma,\tau)=d_\sigma\sigma(d_\tau)/d_{\sigma\tau}\). \(\square\)

The generators in (8) must be a field-valued factor system. Arbitrarily choosing generators of its principal ideals does not guarantee the field cocycle identity: their associativity defect might be a nontrivial unit. This is one reason to retain Noether's refined definition.

Define \(\mathcal P_{\mathrm{ram}}\) to be the subgroup of ideal-valued 2-cocycles consisting of systems \(((a(\sigma,\tau)))\) for which there exists a field-valued factor system \(a\) with that principal-ideal system and with \(A(a)\) split at every ramified place. It is a subgroup because multiplication and inversion of the field cocycles give tensor products and inverse Brauer classes locally, by lesson 5. It contains \(\delta P_L\), the coboundaries of principal-ideal cochains: their field generators give globally split crossed products.

For a normalized family of ideal classes \(c_\sigma\in\operatorname{Cl}(L)\), choose ideal representatives \(\mathfrak c_\sigma\). Say that the family lies in Noether's principal genus if

This includes the requirement that \(c_{\sigma\tau}=c_\sigma\sigma(c_\tau)\). Changing representatives multiplies \(\delta\mathfrak c\) by a member of \(\delta P_L\), so (9) is well-defined. It is stronger than merely saying that each ideal in \(\delta\mathfrak c\) is principal.

Theorem 4.4 (Noether's principal genus theorem, work 41 §2). A family satisfying (9) has the form

Conversely every family (10) satisfies (9).

Proof. Condition (9) gives field cocycle generators \(a\) satisfying (8) and the local splitting hypothesis of Lemma 4.3. That lemma gives \(a=\delta d\). Remove the principal ideals \((d_\sigma)\) from the chosen representatives:

Equation (8) now says \(\delta\mathfrak e=1\). Lemma 3.1 gives \(\mathfrak e_\sigma=\mathfrak b/\sigma(\mathfrak b)\). Passing to classes yields (10), with \(b=[\mathfrak b]\). Conversely choose representatives \(\mathfrak c_\sigma=\mathfrak b/\sigma(\mathfrak b)\). Their ideal factor system is \(1\), generated by the field factor system \(1\), whose algebra splits globally. Thus (9) holds. \(\square\)

This is the second of Noether's three equivalent forms, translated to our fixed left-action convention. Its proof is exactly the two-lemma reduction: remove a principal coboundary using splitting, then solve the ideal cocycle by the ideal sum. The first form describes the same fact through substitutions of the operator cosets by ideal classes, with their factor systems taken modulo the refined subgroup \(\mathcal P_{\mathrm{ram}}\). Conjugation by \(b\) produces the scalar \(b/\sigma(b)\), so (10) says that the prescribed substitution is inner. The third form says that all degree-one crossed representations belonging to this identity factor-system class are equivalent. In degree one, change of basis by \(b\) is again exactly (10). These are statements about multiplicative ideal classes and operator cosets, rather than an additive algebra of ideal classes. Noether explicitly restricts the first form to substitutions of the indicated kind; the ideal class group need not be a maximal commutative subgroup.

5. The later idèle-class formulation

An idèle is a family \(x=(x_w)_w\), \(x_w\in L_w^\times\), such that \(x_w\) is a unit at almost every finite place. Let \(J_L\) be the group of idèles, and \(C_L=J_L/L^\times\) the idèle class group, where \(L^\times\) is embedded diagonally. These are multiplicative \(G\)-modules. Because \(G\) is finite, the group cohomology below is the ordinary finite-group cohomology of these modules; no new continuous-cochain convention is needed.

For this formulation we can give the full reduction to (1). This formulation is due to Artin and Tate.

Lemma 5.1. \(H^1(G,J_L)=1\).

Proof. First fix one place \(v\) of \(K\). Set \(H=G_w\), \(E=L_w\). The product of the groups \(L_{w'}^\times\), for \(w'\mid v\), can be described as the functions \(f:G\to E^\times\) satisfying \(f(gh)=h^{-1}(f(g))\), with action \((\sigma f)(g)=f(\sigma^{-1}g)\). The coordinate at the coset \(gH\) is transported from the completion at \(gw\) back to \(E\) by \(g^{-1}\). This explains both the constraint and the action.

Let \(x_\sigma\) be a 1-cocycle with values in this product. Restriction to \(H\) and evaluation at \(1\) gives a cocycle \(x_h(1)\in E^\times\). Local scalar Hilbert 90, already proved for all finite Galois extensions in lesson 6, gives \(b_0\in E^\times\) with \(x_h(1)=b_0/h(b_0)\). Define

The cocycle law gives

so \(b(gh)=h^{-1}(b(g))\), as required. Applying it instead to \(g^{-1}\sigma\) gives

Thus this product-valued cocycle is \(b/\sigma(b)\). The argument includes archimedean completions: the nontrivial case is \(\mathbb C/\mathbb R\).

Do this for every \(v\), and check the restricted-product condition. Since there are only finitely many \(x_\sigma\), their coordinates are all units outside one finite set of places. Enlarge that set by the ramified places. At an unramified place outside it, the chosen \(b_0\) can be made a unit: multiply it by the appropriate power of a ground-field uniformizer, which has valuation one in the unramified extension and does not change \(b_0/h(b_0)\). Then every \(b(g)\) is a unit too, since every denominator \(x_{g^{-1}}(1)\) is a unit. The resulting family \(b\) is therefore an idèle. It solves the original cocycle at every coordinate, proving the lemma. \(\square\)

Theorem 5.2 (idèle-class principal genus, conditional on (1)). For every finite Galois extension of number fields,

Proof. Let \(c_\sigma\in C_L\) be a normalized cocycle and choose idèle representatives \(j_\sigma\), with \(j_1=1\). Its multiplication defects lie in the diagonal subgroup \(L^\times\):

Associativity, or cancellation of the three cochain factors, makes \(a\) a normalized field-valued 2-cocycle. At a chosen completion \(w\), restriction to \(H=G_w\) writes \(a|_H\) as the coboundary of \(j_h(w)\in L_w^\times\). Therefore its local crossed product splits. Lemma 4.2 says that \(A(a)\) splits at every completion of \(K\). The local–global theorem (1) makes it split globally, and lesson 5 gives \(a=\delta d\) for a field-valued cochain \(d_\sigma\).

The idèles \(e_\sigma=j_\sigma/d_\sigma\) now satisfy \(\delta e=1\). Lemma 5.1 gives \(e_\sigma=b/\sigma(b)\) for an idèle \(b\). Quotienting by \(L^\times\) removes the \(d_\sigma\), so \(c_\sigma=[b]/\sigma([b])\) in \(C_L\). Every cocycle is a coboundary. \(\square\)

The cohomological mechanism is the connecting map

from \(1\to L^\times\to J_L\to C_L\to1\). Lemma 5.1 makes it injective. The local lifts make its image locally zero; (1), combined with the relative-Brauer isomorphism, makes that image globally zero. The preceding proof spells out the connecting map and both removals of coboundaries, so this explanation does not leave an exact-sequence step unproved.

The analogous exact sequence \(1\to P_L\to I_L\to\operatorname{Cl}(L)\to1\) sends a class cocycle to its ideal factor system in \(H^2(G,P_L)\). Lemma 3.1 makes this connecting map injective: if that system is the coboundary of principal ideals, remove those ideals from the representatives and apply the lemma. Noether's refined principal class consists of the systems represented by \(\mathcal P_{\mathrm{ram}}\). Theorem 4.4 says that its intersection with the connecting-map image is zero. This is the precise ideal-class version of the two-step argument.

It does not say that \(H^1(G,\operatorname{Cl}(L))\) always vanishes. For \(L=\mathbb Q(\sqrt{-5})\), the class group is \(\mathbb Z/2\), on which conjugation acts trivially. Sending its nonidentity Galois automorphism to the nonidentity ideal class gives a nontrivial class cocycle; all class coboundaries are trivial. The refined local factor-system condition fails for that cocycle. The idèle class group retains the local multiplicative information lost when one passes directly to ideal classes.

Noether's 1933 paper formulates Theorem 4.4 through ideals, factor systems and operator cosets. Theorem 5.2 is the later idèle-class reading, with the explicit comparison just given. Her 1932 ICM lecture, work 39 §§4–5, already explains why splitting algebras supplies the invariant generalization of cyclic norm statements.

Biquadratic norms and the obstruction group

For a finite Galois extension write

The numerator consists exactly of elements which are norms at every completion. Local preimages may be chosen to be units outside finitely many places by Lemma 4.1, so the local condition gives an actual idèle, not merely an unrestricted product.

Theorem 5.3 (biquadratic norm obstruction). If \(G\simeq C_2\times C_2\), then \(\mathcal K(L/K)\simeq C_2\) when every decomposition group is cyclic, and \(\mathcal K(L/K)=1\) when some decomposition group is \(G\).

Proof. Use the complete-resolution Tate groups and their exact sequences constructed in Brauer groups of local and global fields, “Complete resolutions and fundamental classes”. Applied to \(1\to L^\times\to J_L\to C_L\to1\), exactness gives

Indeed the next map is \(K^\times/N_GL^\times\to J_K/N_GJ_L\), whose kernel is the displayed norm obstruction. The coefficient norm on \(L^\times\) is the field norm, and fixed idèles are precisely \(J_K\).

Theorem 24.6 of that preceding arithmetic lesson gives

At a place \(v\), Proposition 24.3 similarly identifies \(\widehat H^{-1}(G_w,L_w^\times)\) with \(H_2(G_w,\mathbb Z)\). Its subsection on local-to-global compatibility proves compatibility of the local and global fundamental classes under insertion of a local component. Shapiro and the projection formula therefore make the map in (B1) the sum of the homomorphisms

induced by subgroup inclusion. The idèle restricted product causes no additional term. At unramified places the decomposition group is cyclic and the unit module is Tate-acyclic: degree zero is the unit-norm calculation, degree minus one follows by Hilbert 90 and rescaling a primitive by a ground-field uniformizer, and cyclic periodicity supplies every degree. Finite chains and cochain relations thus belong to one finite chart; Shapiro on that chart gives the direct sum of the local groups. These are the unit-tail and finite-chart arguments of Idèles in extensions and their cohomology, applied to the same complete resolution.

It remains to compute the small integral homology groups; no obstruction criterion is assumed. The cyclic periodic resolution has alternate maps \(g-1\) and \(1+g+\cdots+g^{n-1}\). Tensoring with the trivial \(\mathbb Z\)-module makes these maps \(0\) and multiplication by \(n\). In degree two its kernel is zero, so \(H_2(C_n,\mathbb Z)=0\).

For \(G=C_2\times C_2\), tensor the two cyclic free resolutions over \(\mathbb Z\) and take the total differential \(d(x\otimes y)=dx\otimes y+(-1)^{\deg x}x\otimes dy\). This is a free \(\mathbb Z[G]\)-resolution: as complexes of abelian groups, each augmented cyclic resolution splits and contracts onto \(\mathbb Z\). Contracting its positive-degree part in the tensor complex leaves the other resolution, which is exact in positive degrees. After tensoring with the trivial module, the groups in degrees one and two are \(\mathbb Z^2\) and \(\mathbb Z^3\), indexed by bidegrees. The degree-two boundary is

Its kernel is the middle \((1,1)\)-summand. The degree-three boundaries from bidegrees \((2,1)\) and \((1,2)\) have middle coordinates \(2\) and \(-2\); the other two bidegrees give zero. Hence

If all decomposition groups are cyclic, every source of (B3) is zero, and (B1)–(B4) give \(\mathcal K(L/K)\simeq C_2\). If some decomposition group is \(G\), its map in (B3) is the identity. The cokernel is then zero. These exhaust the subgroups of \(C_2\times C_2\), proving both assertions. \(\square\)

In particular, when the obstruction is \(C_2\), a locally norm element outside the global norm subgroup exists. The argument proves that existence; it determines the obstruction group without choosing a numerical representative. The norm theorem for cyclic extensions and the splitting theorem for general crossed products retain exactly their stated scopes.

6. Exercises

Exercise 6.1 (easy). Enumerate the reduced primitive positive definite forms of discriminant \(-20\), compute their genus characters, and identify the principal genus.

Exercise 6.2 (medium). Show that every square class of an imaginary quadratic field lies in the principal genus. Then explain why this alone does not prove equality, and supply the cardinality argument giving the converse.

Exercise 6.3 (medium). Verify the cyclic principal genus theorem for \(\mathbb Q(\sqrt{-5})/\mathbb Q\), using its class group of order two. Check explicitly that the nonprincipal prime class above \(2\) fails the norm-residue condition on its norm generator.

Exercise 6.4 (medium). Hasse's biquadratic example is \(L=\mathbb Q(\sqrt{-3},\sqrt{13})\). Use Theorem 5.3: the norm-obstruction group is nontrivial exactly when every decomposition group is cyclic. Verify that this criterion applies to \(L/\mathbb Q\). Explain why this refutes a norm theorem for all abelian extensions and why it does not refute Theorem 4.4.

Exercise 6.5 (hard). Express Noether's argument in cohomological language using \(1\to P_L\to I_L\to\operatorname{Cl}(L)\to1\). Define the connecting map on representatives, prove its injectivity, and identify the precise local subgroup whose intersection with its image vanishes. Compare the idèle-class connecting map.

7. Solutions

Solution 6.1. Reduced forms satisfy \(3A^2\le20\), hence \(A=1\) or \(2\), with \(B\) even. For \(A=1\), \(|B|\le1\) forces \(B=0\), giving \(C=5\). For \(A=2\), \(B=0\) gives nonintegral \(C=5/2\); \(B=\pm2\) gives \(C=3\), and the boundary convention chooses \(B=2\). Thus the two classes are \((1,0,5)\) and \((2,2,3)\). The prime-discriminant factors are \(-4,5\). The first form represents \(1\), giving signs \((+1,+1)\). The second represents \(3\), giving \(\chi_{-4}(3)=-1\) and \(\chi_5(3)=(3/5)=-1\). There are two genera, each with one class; the first is the principal genus. The class group has two elements, so its square subgroup is exactly that identity class.

Solution 6.2. A genus character is a homomorphism to \(\{\pm1\}\). Hence \(\chi_i(c^2)=\chi_i(c)^2=1\) for each \(i\), proving \(C^2\subseteq\ker\chi\). Characters of a finite abelian group can kill a larger subgroup, so this inclusion has no converse by itself. In our situation the independent prime-discriminant characters make \(C/C^2\to E\) surjective. Conjugation is inversion, and Hilbert 90 converts every conjugation-fixed class into an invariant ideal. Factoring such an ideal shows that \(C[2]\) is generated by the ramified primes; the relation from \((\sqrt d)\), or \((1+i)\) when \(d=-1\), gives \(|C[2]|\le2^{t-1}\). Since \(|C/C^2|=|C[2]|\) and \(|E|=2^{t-1}\), the surjection is bijective. Therefore \(\ker\chi=C^2\). This proves the missing inclusion rather than assuming the principal genus theorem.

Solution 6.3. Conjugation inverts every class; in a group of order two it acts trivially. Thus \((1-\sigma)\operatorname{Cl}(L)\) is the identity subgroup. Theorem 3.2 consequently says that exactly the principal ideal class can have a norm generator satisfying the ramified norm-residue conditions.

Take \(\mathfrak p=(2,1+\sqrt{-5})\). Its norm is \(2\), and \(\mathfrak p^2=(2)\). It is not principal: an integral generator would have norm \(a^2+5b^2=2\), which has no integer solution. It is therefore the nonidentity class. Generators of its norm ideal are \(2\) and \(-2\). At the real place a norm from \(\mathbb C\) must be positive, so \(-2\) fails. At the ramified prime \(5\), \(2\) is a unit. If it were a norm from \(\mathbb Q_5(\sqrt{-5})\), a preimage would be a local unit: in this ramified quadratic extension, the norm valuation equals its normalized extension valuation. The residue norm of a unit is a square in \(\mathbb F_5^\times\), whereas \(2\) is not a square modulo \(5\). Thus \(2\) fails at \(5\). No generator passes all ramified places. This gives an explicit check on the nonprincipal class, while the principal class has generator \(N\beta\) and satisfies (6).

Solution 6.4. The square classes of \(-3\) and \(13\) are independent, so the Galois group is \((\mathbb Z/2)^2\), abelian and noncyclic. The three quadratic subfields have discriminants \(-3\), \(13\), and \(-39\); only \(3\) and \(13\) ramify. At \(3\), \(13\equiv1\pmod3\), so \(3\) splits in \(\mathbb Q(\sqrt{13})\). The decomposition group in \(L\) is consequently contained in the order-two subgroup fixing that subfield. At \(13\), \(-3\equiv10\equiv6^2\pmod{13}\), so \(13\) splits in \(\mathbb Q(\sqrt{-3})\), and its decomposition group likewise has order at most two. All other finite places are unramified, with cyclic decomposition group generated by Frobenius. At the real place the completion is \(\mathbb C\), with an order-two decomposition group. Thus every decomposition group is cyclic.

Theorem 5.3 gives failure of the Hasse norm principle: there exists \(a\in\mathbb Q^\times\) which is a norm at every completion but is not \(N_{L/\mathbb Q}(x)\) for any \(x\in L^\times\). The existence follows from the proved nonzero cokernel (B1), whose order is two; no particular numerical element is needed for this existence assertion. It disproves the proposed extension of the cyclic Hasse norm theorem to all abelian groups. It leaves (1) intact: for a noncyclic extension an element norm condition is not the splitting condition for one cyclic algebra with that extension as its cyclic coefficient field. Noether imposes actual local splitting of a field-valued factor system; Lemma 4.3 and (1) still apply to every Galois group.

Solution 6.5. Represent a class cocycle \(c_\sigma\) by ideals \(\mathfrak c_\sigma\), with \(\mathfrak c_1=\mathcal O_L\). Its defect (4) is a principal-ideal-valued 2-cocycle. If representatives change to \((x_\sigma)\mathfrak c_\sigma\), the defect changes by \(\delta((x_\sigma))\); thus its class in \(H^2(G,P_L)\) is independent of the representatives. Changing the class cocycle by a class coboundary also preserves this class: lift that coboundary to \(\mathfrak b/\sigma(\mathfrak b)\), whose defect is \(1\). These verifications define the connecting map

If \(\kappa([c])=1\), its defect is \(\delta((d_\sigma))\) for principal ideals. The ideals \(\mathfrak c_\sigma/(d_\sigma)\) are a genuine ideal cocycle. Lemma 3.1 makes them \(\mathfrak b/\sigma(\mathfrak b)\), so \([c]=1\). This proves injectivity directly.

Let \(R_{\mathrm{ram}}\subseteq H^2(G,P_L)\) be the subgroup of classes having a representative in \(\mathcal P_{\mathrm{ram}}\). This is well-defined under principal-ideal coboundaries because those are generated by globally split field cocycles. If \(\kappa([c])\in R_{\mathrm{ram}}\), multiply a chosen ideal defect by the inverse of the principal-ideal coboundary needed to make it a representative in \(\mathcal P_{\mathrm{ram}}\); equivalently change its ideal representatives. There are now field cocycle generators \(a\) splitting at the ramified places. Lemma 4.3 makes \(a=\delta d\), hence \(\kappa([c])=1\), and injectivity gives \([c]=1\). In symbols,

For the idèle sequence the representatives are \(j_\sigma\), and their defects are in \(L^\times\). Lemma 5.1 makes the connecting map injective. Every restricted local defect is a coboundary of a completion-valued cochain, so every associated local algebra splits. Theorem (1) makes the global cocycle a coboundary, proving the entire group \(H^1(G,C_L)\) trivial. The different local information in the two coefficient groups explains why this does not assert the same vanishing for \(\operatorname{Cl}(L)\).

Sources and further reading

• [Noether, work 41] Emmy Noether, Der Hauptgeschlechtssatz für relativ-galoissche Zahlkörper, free English edition, Mathematische Annalen 108 (1933), 411–419, especially §2.1 for the refined class division, §2.2 for the three forms, and §2.3 for the two lemmas and their combination. The English collected edition provides a companion text. Section 4 independently writes the reduction in the left-action convention of this course.
• [Noether, work 39] Emmy Noether, Hyperkomplexe Systeme in ihren Beziehungen zur kommutativen Algebra und zur Zahlentheorie, free original proceedings, Verhandlungen des Internationalen Mathematiker-Kongresses Zürich 1 (1932), 189–194, §§4–5. The lecture explains cyclic norms and the invariant splitting-algebra formulation.
• [Brauer–Hasse–Noether, 1932] Richard Brauer, Helmut Hasse and Emmy Noether, Beweis eines Hauptsatzes in der Theorie der Algebren, free English collected edition, Journal für die reine und angewandte Mathematik 167 (1932), 399–404. The local–global input (1) is used in the modern form of [Milne CFT, Chapter VIII, Theorem 4.2].
• [Milne CFT] J. S. Milne, Class Field Theory, course notes, Chapter VIII: Theorem 3.1 (the Hasse norm theorem for cyclic extensions; it does not extend to arbitrary abelian extensions) and Theorem 4.2 (the local–global exact sequence of Brauer groups). The preceding programme lessons linked in section 1 prove these global theorems; their applications here retain the cyclic and local hypotheses.
• [Rome] Nick Rome, The Hasse Norm Principle for Biquadratic Extensions, Lemma 2.1 and its proof, pp.3–4 of arXiv:1707.00412v2. Theorem 5.3 supplies the norm-obstruction proof used in Exercise 6.4; its solution computes the decomposition groups of the indicated field.
• [Terada] Fumiyuki Terada, On the principal genus theorem concerning the abelian extensions, Tohoku Mathematical Journal, Second Series 4 (1952), 141–152: the introduction, p.141, gives Hasse's cyclic ray formulation; §1, equation (11), p.143, specifies its conductor and genus modulus. Theorem 3.4 proves the ray assertion in the conventions used here. The original article fixes the modulus conventions used in section 3.
• [Lemmermeyer] Franz Lemmermeyer, The Development of the Principal Genus Theorem, arXiv:math/0207306, p.23. Section 3 explains the conductor qualification needed when using its abbreviated ray-class statement.

The arithmetic proofs and the quadratic-form background are the programme readings linked in section 1. Crossed-product factor systems connect the representation theory developed here to these norm and ideal-class calculations.

Scope and prerequisites

The arithmetic proofs used here are the precisely linked cyclic norm, Brauer local–global, quadratic-form, reciprocity, local ramification and class-formation results of the preceding programme. Section 1 specifies the acyclic route from the algebra of lessons 4–5 through arithmetic to this lesson. The imaginary quadratic character kernel, cyclic ideal and ray assertions, local norm calculation beyond the conductor, Noether's refined ideal-class theorem, idèle reductions and biquadratic obstruction are proved here from those inputs, with all five exercise solutions. Ordinary and narrow groups, finite and infinite places, and cyclic and arbitrary finite Galois extensions keep their distinct conditions throughout.