The narrow relation between the ramified primes of a quadratic field #
Let K = ℚ(√d) be a quadratic number field, presented by θ : 𝓞 K with
minpoly ℤ θ = X ^ 2 - d and Algebra.adjoin ℚ {θ} = ⊤, with d squarefree and 1 < |d|, and
let 𝔭_p be the prime of 𝓞 K above a ramified rational prime p. The narrow classes [𝔭_p]⁺
are involutions in the narrow class group Cl⁺(K), so a priori they span at most 2 ^ t classes
with t the number of ramified primes. This file produces one relation between them, cutting
the bound to 2 ^ (t - 1):
∃ S ≠ ∅, ∏_{p ∈ S} [𝔭_p]⁺ = 1.
In the ordinary class group the relation is the single explicit identity ∏_{p ∣ d} [𝔭_p] = 1,
coming from (θ) = ∏_{p ∣ d} 𝔭_p (TauCeti.Multiquadratic.prod_classGroupMk0_eq_one). Narrowly
that identity survives only when (θ) has a totally positive generator, and for a real quadratic
field it often does not: for K = ℚ(√3) the narrow class of 𝔭_3 = (√3) is nontrivial, and the
relation is instead [𝔭_2]⁺[𝔭_3]⁺ = 1; for K = ℚ(√7) it is [𝔭_2]⁺ = 1, since 3 + √7 is
totally positive of norm 2. There is no uniform choice of S, and the proof splits accordingly.
- If some unit
umakesθutotally positive up to sign,Sis the set of prime factors ofd. This covers every imaginary quadratic field, where total positivity is vacuous, and every real quadratic field admitting a unit of norm-1. - Otherwise every unit has norm
1, so every unit is±a totally positive one; since a field with a real place has unit rank1, where the squares have index2 ^ (rank + 1) = 4(NumberField.units_sq_index_eq), some totally positive unitεis not a square. Hilbert 90 turnsεintoz ≠ 0withσz = εz; the ideal(z)is then ambiguous, so it is a positive rational integer times a product of distinct ramified primes, and dividing out that rational factor leaves∏_{p ∈ S} 𝔭_p = (γ)withγ/σγ = ε⁻¹totally positive. WereSempty,γwould be a unitvwithσv = εv, forcingε = (v⁻¹)², a square.
Both branches are archimedean where the ordinary theory is not: what replaces the ordinary
argument's use of total complexity is total positivity, exactly as in the narrow Hilbert-90 descent
of TauCeti.NumberTheory.NumberField.Quadratic.Conjugation.Ambiguous.Narrow.
The classical source is F. Lemmermeyer, Reciprocity Laws: From Euler to Eisenstein, §2.2, and
D. A. Cox, Primes of the Form x² + ny², §6.A, where this is the "first inequality"
#Cl⁺/(Cl⁺)² ≤ 2 ^ (t - 1) of the ambiguous class number formula.
Main results #
In the namespace TauCeti.Multiquadratic:
exists_nonempty_prod_narrowMk0_eq_one_of_unit: the Hilbert-90 construction of a relation from such a unit.exists_nonempty_prod_narrowMk0_eq_one: the relation, for a quadratic field of either signature.natCard_closure_image_narrowMk0_le: hence the narrow classes of the ramified primes generate a subgroup of order at most2 ^ (t - 1).
The relation #
A totally positive unit that is not a conjugation ratio produces a relation. Let ε be a
totally positive unit of norm one such that σv = εv for no unit v. Hilbert 90 produces z ≠ 0
with σz = εz, so the ideal (z) is ambiguous; the structure theorem writes it as a positive
rational integer times a product ∏_{p ∈ S} 𝔭_p of distinct ramified primes, and stripping the
rational factor leaves ∏_{p ∈ S} 𝔭_p = (γ) with γ/σγ = ε⁻¹ totally positive. Hence γ or -γ
is totally positive and the product has trivial narrow class. The set S is nonempty: were it
empty, γ would be a unit v with σv = εv.
An explicit relation of narrow genus theory. For K = ℚ(√d) with d squarefree and
1 < |d|, some nonempty set S of ramified primes has ∏_{p ∈ S} [𝔭_p]⁺ = 1 in Cl⁺(K).
Which set works depends on K. If some unit u makes θu totally positive up to sign — in
particular whenever K is imaginary, where the condition is vacuous, and whenever some unit has
norm -1 — then S is the set of prime factors of d, because (θ) = ∏_{p ∣ d} 𝔭_p. Otherwise
K is real with every unit of norm one, and
exists_nonempty_prod_narrowMk0_eq_one_of_unit builds S by Hilbert 90 from a totally
positive unit that is not a square (exists_isTotallyPositive_notMem_square).
Unlike the ordinary relation prod_classGroupMk0_eq_one, which is the single identity
∏_{p ∣ d} [𝔭_p] = 1, no uniform choice of S is available: for K = ℚ(√3) the relation is
[𝔭_2]⁺[𝔭_3]⁺ = 1 with [𝔭_3]⁺ ≠ 1, whereas for K = ℚ(√7) it is [𝔭_2]⁺ = 1. The hypothesis
1 < |d| excludes d = -1, where the radicand has no prime factor and the first branch would
produce the empty set.
The narrow classes of the ramified primes generate a subgroup of order at most
2 ^ (t - 1). Let s be the finite set of ramified primes of K = ℚ(√d), with Q p the prime
of 𝓞 K above p. Then the subgroup of Cl⁺(K) generated by their narrow classes has at most
2 ^ (t - 1) elements, where t = #s.
The classes are involutions (NarrowClassGroup.mk0_sq_eq_one_of_mem_ramifiedPrimes), so the
subgroup they generate is the set of sub-products; the relation
exists_nonempty_prod_narrowMk0_eq_one removes one generator.