Louboutin's Conjecture 6.1 in an abstract biquadratic model #
Model (Louboutin.Conj61.V4Data O): O a commutative domain of characteristic 0 (think 𝓞_K),
four distinct elements x 0, …, x 3 (the conjugates of α = x 0), three ring endomorphisms
τ₁, τ₂, τ₃ acting on them by the labelling τ_i x_k = x_{i xor k}, and fix: an element fixed
by τ₁, τ₂, τ₃ is an integer (for 𝓞_K with Gal(K/ℚ) ≅ V₄, 𝓞_K^G = ℤ). Zα is the
ℤ-span of 1, α, α², α³ (= ℤ[α], Zα_mul_mem); "ℤ[α] Galois-invariant" means that every
conjugate lies in Zα.
The route avoids content ideals over 𝓞_K: for each j, r_j² and r_j ΔQ_j are invariant,
hence integers; dividing by their gcd gives r_j = n_j ϑ_j, ΔQ_j = t_j ϑ_j with ϑ_j ∈ O
(Bézout), Θ = ϑ₁ϑ₂ϑ₃ is an integer θ, and P(X) = θ ∏(n_j X + t_j) with a primitive
product (Gauss's lemma, IsPrimitive.mul).
Main results #
Louboutin.Conj61.V4Data.thm2: an integer dividingT, U, V, Wdivides everyN_jk.Louboutin.Conj61.V4Data.gcd_TUVW:gcd(T, U, V, W) = |θ|.Louboutin.Conj61.V4Data.conj61: Conjecture 6.1,ℤ[α]is invariant iffΔ = VanddividesT, U, V, W.Louboutin.Conj61.V4Data.exists_ints:T, U, V, W, Δare integers,Δ ≠ 0andΔis even.Louboutin.Conj61.V4Data.cor_weak,Louboutin.Conj61.V4Data.conj61_weak: invariance iffΔ ∣ T, U, V ± W.
References #
- S. R. Louboutin, On the order generated by the conjugates of an algebraic integer, in: Algebra, Logic and Number Theory, Banach Center Publ. 121 (2020), 107–124, doi:10.4064/bc121-10.
- S. R. Louboutin, When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures, Czechoslovak Math. J. 70 (2020), 905–919, doi:10.21136/CMJ.2020.0019-19.
- C. Marín, Orders generated by conjugates of Galois quartic integers, preprint (2026).
Determinants (Lemma 1.3) #
Lemma 1.3: if column l has conjugates E i l = Σ_k c_lk x_iᵏ (c integral), then det E = Vand · det c.
Integer bookkeeping (the ℤ[X] Gauss step) #
Gauss for primitive polynomials over ℤ: if g divides every coefficient of θ·∏_j (n_j X + t_j) with gcd(n_j, t_j) = 1, then g ∣ θ.
The model #
V4 data: conjugates x, the three nontrivial automorphisms in the XOR labelling, and
"invariant ⇒ integer".
- x : Fin 4 → O
- inj : Function.Injective self.x
Instances For
Saturation: O ∩ ℚ = ℤ in the form g·y = n ⇒ g ∣ n.
Integrality of the symmetric data #
ℤ[α] #
Necessity (Theorem 1): invariant ⇒ Vand ∣ T, U, V, W #
Sufficiency (Theorem 2 + Cramer), via the pairs (r_j, ΔQ_j) #
Theorem 2 (proof of Theorem A), unconditional: any integer g dividing T, U, V and W divides the twelve Cramer numerators N_jk (which are integers).
Θ-factorisation (the S4 prefix of thm2, stated on its own): coprime pairs
(n_j, t_j), elements ϑ_j ∈ O with r_j = n_j ϑ_j, ΔQ_j = t_j ϑ_j, and an integer θ = ϑ₁ϑ₂ϑ₃
with P(X) = θ·∏(n_j X + t_j): U, V, W, T are θ times the coefficients of the primitive
polynomial ∏(n_j X + t_j).
Content of P (proof of Theorem A): θ = ϑ₁ϑ₂ϑ₃ (ϑ_j from split_pair, r_j = n_j ϑ_j,
ΔQ_j = t_j ϑ_j with gcd(n_j, t_j) = 1) divides T, U, V, W, and every common divisor of
T, U, V, W divides θ: θ is a gcd of (T, U, V, W).
gcd(T, U, V, W) = |Θ| literally: Int.gcd (Int.gcd t u) (Int.gcd v w) = |θ|, with
θ = ϑ₁ϑ₂ϑ₃ as in gcd_eq_abs_theta.
Conjecture 6.1 (Louboutin), in the V4 model: ℤ[α] is Galois-invariant (contains all conjugates) iff Δ = Vand divides T, U, V and W.
Corollary cor:weak: Δ ∣ T, U, V ± W already forces Δ ∣ V, W #
Route (the note's, without g ∣ Δ): cross_prod at r_j = n_jϑ_j, ΔQ_j = t_jϑ_j gives
θ²·d₁₂d₂₃d₃₁ = Δ·θ·n₁n₂n₃ (theta_minors). Write Δ = hG, θ = θ'G with gcd(h, θ') = 1; the
hypothesis gives h ∣ f₃, f₀, f₂ ± f₁ (f the coefficients of the primitive ∏(n_jX + t_j)), and
h ∣ d₁₂d₂₃d₃₁. If h were even, the mod-2 analysis of the three coprime pairs (par_core) makes
d₁₂d₂₃d₃₁ odd: contradiction. So h is odd, h ∣ 2f₂, 2f₁ gives h ∣ f₂, f₁, and Δ ∣ V, W.
mod-2 core: three nonzero pairs (n_j, t_j) over 𝔽₂ with n₁n₂n₃ = 0, t₁t₂t₃ = 0 and f₂ + f₁ = 0 are the three points of ℙ¹(𝔽₂), so every minor d_ij is nonzero.
Integer form of par_core.
Θ and the minors: with the data of theta_factor, θ²·d₁₂d₂₃d₃₁ = Δ·θ·n₁n₂n₃,
d_ij = n_i t_j − n_j t_i (cross_prod at r_j = n_jϑ_j, ΔQ_j = t_jϑ_j).
Corollary cor:weak (in the V4 model): if Δ = Vand divides T, U, V + W and V − W, then
Δ divides T, U, V and W.
Cor. cor:weak + Conj. 6.1: ℤ[α] is Galois-invariant iff Δ divides T, U, V + W and
V − W (Louboutin's necessary condition, Prop. 6.1, is sufficient).