The narrow-versus-ordinary defect of a quadratic field #
For a number field K the sequence Kˣ → Cl⁺(K) → Cl(K) → 1 is exact, so the defect between the
narrow and the ordinary class group is the image of the principal-class map mkPrincipal. That
image is a quotient of the group of sign patterns of Kˣ at the real places, modulo the global
sign, since (x) and (-x) are the same ideal. A quadratic field has at most two real places, so
the defect has at most two elements.
The quantitative statement proved here is sharper than a place count, and needs no case split on
the signature. Let σ be quadratic conjugation and let x ∈ Kˣ. The ratio r = x / σx has
σ r = r⁻¹, so r / σ r = r² is totally positive, and
NumberField.isTotallyPositive_or_isTotallyPositive_neg_of_isTotallyPositive_div_quadraticConj
puts r or -r on the totally positive side. In the first case the same lemma applied to x
makes x or -x totally positive, so (x) is narrowly trivial. In the second case the generator
θ absorbs the discrepancy: σθ = -θ gives (θx) / σ(θx) = -r, so θx or -θx is totally
positive and [x]⁺ = [θ]⁺. Hence every principal narrow class is 1 or [θ]⁺.
This bounds by one the amount by which the ordinary 2-rank of a quadratic field can fall short
of the narrow 2-rank t - 1 computed by genus theory: for a real field the drop does happen,
as ℚ(√3) shows.
Whether the defect is trivial is decided by the units. A unit of norm -1 scales every nonzero
element to a totally positive one
(NumberField.exists_unit_isTotallyPositive_smul_of_norm_eq_neg_one), so every principal narrow
class is trivial and Cl⁺(K) → Cl(K) is injective; the field is then
automatically real. Conversely, for a real field (0 < d), injectivity makes the narrow class of
(θ) trivial, so some v · θ is totally positive and has positive norm N(v) · (-d), forcing
N(v) = -1. This is the classical criterion h⁺ = h ↔ N(ε) = -1 for the fundamental unit ε. A
solution of the negative Pell equation b² - d a² = -1 supplies such a unit
(NumberField.exists_norm_eq_neg_one_of_sq_sub_mul_sq_eq_neg_one): for d = 2, a = b = 1 gives
1 + √2. For an imaginary field there is no unit of norm -1, and the two class groups agree for
the unrelated reason that positivity is vacuous
(NumberField.NarrowClassGroup.toClassGroup_injective).
Main results #
NumberField.mkPrincipal_eq_one_or_eq_mkPrincipal_gen: a principal narrow class of a quadratic field is trivial or the narrow class of the generator.NumberField.mkPrincipal_eq_mkPrincipal_gen_of_norm_neg: a principal ideal with a generator of negative norm has the narrow class of the generator.NumberField.card_ker_toClassGroup_le_two: the kernel ofCl⁺(K) → Cl(K)has at most two elements.NumberField.card_narrowClassGroup_le_two_mul_card_classGroup:h⁺(K) ≤ 2 h(K).NumberField.NarrowClassGroup.toClassGroup_injective_of_norm_eq_neg_one: a unit of norm-1makesCl⁺(K) → Cl(K)injective.NumberField.NarrowClassGroup.toClassGroup_injective_iff_exists_norm_eq_neg_one: for0 < dthe converse holds too.NumberField.NarrowClassGroup.card_eq_card_classGroup_iff_exists_norm_eq_neg_one: for0 < dthe narrow class number equals the class number exactly when some unit has norm-1.
References #
- D. A. Cox, Primes of the Form x² + ny², §6.A, and F. Lemmermeyer, Reciprocity Laws: From
Euler to Eisenstein, §2.2, for the narrow class group of a quadratic field and its comparison
with the ordinary class group,
h⁺(K) ∈ {h(K), 2 h(K)}, which is the boundcard_narrowClassGroup_le_two_mul_card_classGroupproved here.
The principal narrow classes of a quadratic field are 1 and [θ]⁺. For K = ℚ(√d)
presented by θ and any x : Kˣ, the narrow class of the principal ideal (x) is trivial or
equal to the narrow class of (θ).
The ratio r = x / σx satisfies σ r = r⁻¹, so r / σ r = r² is totally positive and hence r
or -r is. If r is, then x or -x is totally positive and (x) is narrowly trivial. If -r
is, the same applies to θx, because σθ = -θ turns (θx) / σ(θx) into -r.
A generator of negative norm has the narrow class of (θ). For K = ℚ(√d) presented by
θ and x : Kˣ with N(x) < 0, the narrow class of (x) is that of (θ), even when both are
trivial.
The narrow class group of a quadratic field exceeds the ordinary one by at most a factor
of two. By exactness the kernel of Cl⁺(K) → Cl(K) is the image of the principal-class map.
Choosing a presentation θ of K (exists_minpoly_eq_X_sq_sub_C_and_adjoin_eq_top),
mkPrincipal_eq_one_or_eq_mkPrincipal_gen confines that image to the subgroup generated by the
narrow class of (θ), which is 2-torsion.
The narrow class number of a quadratic field is at most twice the class number.
A unit of norm -1 makes the narrow class group the ordinary one. If some unit of 𝓞 K
has norm -1 then every principal narrow class is trivial, so forgetting positivity
Cl⁺(K) → Cl(K) is injective and the two class groups agree. This is the substantial direction of
the classical criterion h⁺ = h ↔ N(ε) = -1; the field is automatically real.
The narrow and ordinary class groups of a real quadratic field agree exactly when some unit
has norm -1. For K = ℚ(√d) with 0 < d, forgetting positivity Cl⁺(K) → Cl(K) is injective
if and only if some unit of 𝓞 K has norm -1. The positivity hypothesis is needed only for the
direction producing such a unit.
The narrow class number equals the class number exactly when some unit has norm -1. The
class-number form of toClassGroup_injective_iff_exists_norm_eq_neg_one, which is how the
classical criterion h⁺ = h ↔ N(ε) = -1 is usually stated.