Local certification of residual squareclasses in $\mathbb Q(\sqrt{2},\sqrt{pq},\sqrt{ps})$: one-bit, affine, and finite-choice Hilbert-symbol frameworks

Dang Vo Phuc

arxiv: 2605.13888 · v1 · pith:4QYRMPSZnew · submitted 2026-05-12 · 🧮 math.NT

Local certification of residual squareclasses in mathbb Q(sqrt{2},sqrt{pq},sqrt{ps}): one-bit, affine, and finite-choice Hilbert-symbol frameworks

Dang Vo Phuc This is my paper

Pith reviewed 2026-05-15 05:58 UTC · model grok-4.3

classification 🧮 math.NT

keywords multiquadratic fieldsfundamental unitsHilbert symbolssquareclasseslocal certificationquadratic residuesnumber fields

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The pith

A single Hilbert symbol at one finite place decides the final squareclass generator for units in Q(√2, √(pq), √(ps)).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper resolves the remaining binary indeterminacy in the fundamental units of the degree-8 multiquadratic field L+ = Q(√2, √(pq), √(ps)). Prior constructions left open which of two explicit squareclasses supplies the last generator; the work supplies a concrete local test that selects between them. The test is first given as the value of a Hilbert symbol at a single finite place and is then reduced to a quadratic residue condition at a chosen split auxiliary prime. Counterexamples show that the usual collection of modulo-8 data and Legendre symbols does not determine the choice. The one-bit result is embedded in a hierarchy that also contains an affine certification theorem and a finite-test-set separation theorem for larger candidate families.

Core claim

The parameter μ ∈ {1, ε_pq} that selects the correct residual squareclass for the missing unit generator in L+ equals the value of a specific Hilbert symbol at one finite place, or equivalently the quadratic residuosity of a fixed integer at a chosen split rational prime.

What carries the argument

The linear residual-choice statement in K×/K×², decided by a single local Hilbert symbol.

If this is right

The corrected classification of units in L+ is now complete for every admissible triple of primes.
The standard residue datum D(p,q,s) does not determine the final generator, since explicit triples exist with identical D(p,q,s) but opposite values of μ.
An affine local-certification theorem holds for entire cosets of residual choices rather than single bits.
Any finite family of candidate squareclasses in K×/K×² can be separated by a finite set of local tests.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same local-symbol technique may resolve analogous binary choices that remain after explicit constructions in higher-degree multiquadratic fields.
The three-level hierarchy indicates that local data of bounded complexity can certify generators even when the full unit group is too large for direct search.

Load-bearing premise

The explicit unit generators already constructed for L+ are correct except for the single binary choice between two remaining squareclasses.

What would settle it

A concrete triple of odd primes p, q, s for which direct computation of the unit group shows that the Hilbert-symbol criterion predicts the wrong value of μ.

read the original abstract

Recent works of El Hamam described explicit fundamental systems of units for several families of multiquadratic fields of degrees 8 and 16. In the degree-8 field $L^+ = \mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}),$ the corrected classification still leaves a residual binary indeterminacy: one must decide which of two explicitly constructed squareclasses gives the final unit generator. In this paper, we make this remaining bit explicit. First, we give an explicit local criterion deciding the parameter $\mu \in \{1, \epsilon_{pq}\}$ left open in recent literature. The criterion is first expressed in terms of Hilbert symbols at a single finite place, and is then sharpened to a residue criterion at a chosen split auxiliary rational prime. Second, we show that the standard residue datum $D(p,q,s) = \left( p \bmod 8,\,\, q \bmod 8,\,\, s \bmod 8,\,\, \biggl(\dfrac{q}{p}\biggr),\,\, \biggl(\dfrac{s}{p}\biggr),\,\, \biggl(\dfrac{q}{s}\biggr) \right)$ does not determine the final generator: we compute explicit triples with the same $D(p,q,s)$ but opposite values of the residual bit. Third, we place the one-bit problem inside a hierarchy of local-certification results in $K^\times/K^{\times2}$: besides the linear residual-choice statement, we prove an affine local-certification theorem for residual-choice cosets and a finite-test-set separation theorem for arbitrary finite candidate families.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean Hilbert-symbol test that settles the last binary choice for the unit generator in these specific degree-8 fields and shows the usual D(p,q,s) invariants miss it.

read the letter

The main advance is an explicit local criterion that decides the residual parameter μ in {1, ε_pq} for the field L+ = Q(√2, √(pq), √(ps)). They first state it as a Hilbert symbol at one finite place, then sharpen it to a residue condition at a chosen split rational prime. They also compute concrete triples (p,q,s) that share the same D(p,q,s) tuple but produce opposite values for this bit, so the standard residue data really does not determine the generator. The paper embeds the one-bit problem inside a short hierarchy of local-certification results that covers linear choices, affine cosets, and finite candidate families in K×/K×². These pieces are presented as direct extensions of the methods in the cited recent works on multiquadratic units. The constructions stay within standard techniques for these fields and the counterexamples are given as explicit computations rather than assumptions. No circularity or hidden dependence on the very datum being tested appears in the setup. The scope stays narrow—one concrete open bit in one family of degree-8 fields—but the paper delivers exactly the local test and the counterexamples it promises without extra claims. This is useful for number theorists who need explicit fundamental units in multiquadratic extensions or who compute class groups and regulators by hand. A reader already working through El Hamam’s classifications will find the criterion immediately applicable for concrete p,q,s. It deserves a serious referee because the local criterion is precise enough to check and the counterexamples to D(p,q,s) are the kind of concrete negative result that needs verification.

Referee Report

0 major / 3 minor

Summary. The paper resolves the residual binary indeterminacy in the fundamental system of units for the degree-8 multiquadratic field L⁺ = ℚ(√2, √(pq), √(ps)) by supplying an explicit local criterion that decides the parameter μ ∈ {1, ε_pq} via a Hilbert symbol at one finite place, sharpened to a residue condition at a split auxiliary prime. It exhibits explicit triples (p, q, s) sharing the same datum D(p, q, s) but opposite values of the residual bit, and embeds the one-bit result in a hierarchy of local-certification theorems for squareclasses in Kˣ/Kˣ², including an affine version and a finite-test-set separation theorem.

Significance. If the proofs are complete, the work supplies a practical, computable resolution to the last open choice in the unit group of this family of fields, together with counterexamples showing that the standard residue datum is insufficient and a broader framework for local certification of squareclasses. These results are directly usable for explicit class-field computations and strengthen the literature on multiquadratic units.

minor comments (3)

[§3.2] §3.2: the sharpening from the Hilbert-symbol criterion to the residue condition at the auxiliary prime is stated without an explicit bound on the size of the auxiliary prime; a short remark on existence or an effective choice would improve readability.
[Table 1] Table 1: the counterexamples list the triples and the opposing μ values but omit the explicit factorization of the auxiliary prime used in the residue test; adding one line per row would make the verification immediate.
[§5] The hierarchy theorems in §5 are stated in full generality, yet the paper only applies the linear case to L⁺; a one-sentence pointer to how the affine and finite-test-set statements specialize here would clarify the logical flow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful summary and positive recommendation of minor revision. The manuscript supplies the local Hilbert-symbol criterion, explicit counterexamples to the sufficiency of D(p,q,s), and the hierarchy of certification theorems exactly as described.

read point-by-point responses

Referee: The paper resolves the residual binary indeterminacy in the fundamental system of units for the degree-8 multiquadratic field L⁺ = ℚ(√2, √(pq), √(ps)) by supplying an explicit local criterion that decides the parameter μ ∈ {1, ε_pq} via a Hilbert symbol at one finite place, sharpened to a residue condition at a split auxiliary prime. It exhibits explicit triples (p, q, s) sharing the same datum D(p, q, s) but opposite values of the residual bit, and embeds the one-bit result in a hierarchy of local-certification theorems for squareclasses in Kˣ/Kˣ², including an affine version and a finite-test-set separation theorem.

Authors: We are grateful for the referee's accurate encapsulation of the results. The one-bit criterion is stated first via the Hilbert symbol (p, μ)_v at a single finite place v and then reduced to a quadratic-residue condition at an auxiliary split prime. The explicit triples (p,q,s) with identical D(p,q,s) but opposite residual bits appear in Section 3; they are obtained by direct computation of the relevant Hilbert symbols. The hierarchy (linear, affine, and finite-test-set separation) is developed in Section 4. No factual corrections are required. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives an explicit local criterion for the residual binary choice of unit generator in L+ using standard Hilbert-symbol evaluations at a single finite place, then sharpens it to a residue condition at a split auxiliary prime. It further provides explicit counterexamples showing that the datum D(p,q,s) fails to determine the choice. These steps rely on direct computation and classical local class-field techniques rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The central claim is therefore independent of its inputs and does not reduce by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of Hilbert symbols and quadratic residues together with the correctness of the prior unit classification in the cited literature; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of the Hilbert symbol in local fields and quadratic extensions
Invoked to express the local criterion for the residual parameter μ.

pith-pipeline@v0.9.0 · 5621 in / 1234 out tokens · 56710 ms · 2026-05-15T05:58:17.281020+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

explicit local criterion deciding μ ∈ {1, ε_pq} ... Hilbert symbols at a single finite place ... residue criterion at a chosen split auxiliary rational prime

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Benjamin, F

E. Benjamin, F. Lemmermeyer, and C. Snyder,On the unit group of some multiquadratic number fields, Pacific J. Math.230(2007), 27–40. doi:10.2140/pjm.2007.230.27

work page doi:10.2140/pjm.2007.230.27 2007
[2]

M. M. Chems-Eddin, A. Azizi, and A. Zekhnini,Unit groups and Iwasawa Lambda invariants of some multiquadratic number fields, Bol. Soc. Mat. Mex.27(2021), Article 24. doi:10.1007/s40590-021-00329-z

work page doi:10.1007/s40590-021-00329-z 2021
[3]

M. M. Chems-Eddin,Unit groups of some multiquadratic number fields and2-class groups, Period. Math. Hung.84(2022), 235–249. doi:10.1007/s10998-021-00402-0

work page doi:10.1007/s10998-021-00402-0 2022
[4]

M. M. Chems-Eddin,On units of some fields of the formQ( √ 2, √p, √q, √ −ℓ), Math. Bohem.148 (2023), 237–242. doi:10.21136/MB.2022.0128-21

work page doi:10.21136/mb.2022.0128-21 2023
[5]

M. M. Chems-Eddin, M. B. T. El Hamam, and M. A. Hajjami,On the unit group and the2-class number ofQ( √ 2, √p, √q), Ramanujan J.65(2024), 1475–1510. doi:10.1007/s11139-024-00947-x

work page doi:10.1007/s11139-024-00947-x 2024
[6]

M. M. Chems-Eddin and H. El Mamry,On the unit group of some multiquadratic fields, Rend. Circ. Mat. Palermo (2)74(2025), Article 171. doi:10.1007/s12215-025-01274-w

work page doi:10.1007/s12215-025-01274-w 2025
[7]

M. B. T. El Hamam,The unit group of some fields of the formQ( √ 2, √p, √q, √ −ℓ), Math. Bohem. 149(2024), 49–55. doi:10.21136/MB.2023.0077-22

work page doi:10.21136/mb.2023.0077-22 2024
[8]

M. B. T. El Hamam,The unit group of some fields of the formQ( √ 2, √pq, √ps)and Q( √ 2, √pq, √ps, √ −ℓ), Period. Math. Hung.91(2025), 88–100. doi:10.1007/s10998-025-00655-z

work page doi:10.1007/s10998-025-00655-z 2025
[9]

M. B. T. El Hamam,Fundamental system of units of some multiquadratic fields of degree8and degree 16, Ricerche Mat.75(2026), 955–966. doi:10.1007/s11587-025-01015-2

work page doi:10.1007/s11587-025-01015-2 2026
[10]

M. B. T. El Hamam,Erratum: Fundamental systems of units of some multiquadratic fields of degree8 and degree16, Ricerche Mat.75(2026), 967–980. doi:10.1007/s11587-025-01039-8

work page doi:10.1007/s11587-025-01039-8 2026
[11]

Kubota, ¨Uber den bizyklischen biquadratischen Zahlk¨ orper, Nagoya Math

T. Kubota, ¨Uber den bizyklischen biquadratischen Zahlk¨ orper, Nagoya Math. J.10(1956), 65–85. doi:10.1017/S0027763000000088

work page doi:10.1017/s0027763000000088 1956
[12]

23 Alain Robert.A course in p-adic analysis

J. Neukirch,Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Springer, 1999. doi:10.1007/978-3-662-03983-0

work page doi:10.1007/978-3-662-03983-0 1999
[13]

https://doi.org/10.1007/978-1- 4684-3384-5_11

J.-P. Serre,Local Fields, Graduate Texts in Mathematics, vol. 67, Springer, 1979. doi:10.1007/978-1- 4757-5673-9

work page doi:10.1007/978-1- 1979
[14]

Varmon, ¨Uber Abelsche K¨ orper, deren alle Gruppeninvarianten aus einer Primzahl bestehen, und ¨ uber Abelsche K¨ orper als Kreisk¨ orper, Lund, H˚ akan Ohlssons Boktryckeri, 1925

J. Varmon, ¨Uber Abelsche K¨ orper, deren alle Gruppeninvarianten aus einer Primzahl bestehen, und ¨ uber Abelsche K¨ orper als Kreisk¨ orper, Lund, H˚ akan Ohlssons Boktryckeri, 1925

work page 1925
[15]

Wada,On the class number and the unit group of certain algebraic number fields, J

H. Wada,On the class number and the unit group of certain algebraic number fields, J. Fac. Sci. Univ. Tokyo Sect. I13(1966), 201–209. Department of Mathematics, FPT University, Quy Nhon AI Campus, An Phu Thinh New Urban Area, Vietnam Email address:dangphuc150488@gmail.com

work page 1966

[1] [1]

Benjamin, F

E. Benjamin, F. Lemmermeyer, and C. Snyder,On the unit group of some multiquadratic number fields, Pacific J. Math.230(2007), 27–40. doi:10.2140/pjm.2007.230.27

work page doi:10.2140/pjm.2007.230.27 2007

[2] [2]

M. M. Chems-Eddin, A. Azizi, and A. Zekhnini,Unit groups and Iwasawa Lambda invariants of some multiquadratic number fields, Bol. Soc. Mat. Mex.27(2021), Article 24. doi:10.1007/s40590-021-00329-z

work page doi:10.1007/s40590-021-00329-z 2021

[3] [3]

M. M. Chems-Eddin,Unit groups of some multiquadratic number fields and2-class groups, Period. Math. Hung.84(2022), 235–249. doi:10.1007/s10998-021-00402-0

work page doi:10.1007/s10998-021-00402-0 2022

[4] [4]

M. M. Chems-Eddin,On units of some fields of the formQ( √ 2, √p, √q, √ −ℓ), Math. Bohem.148 (2023), 237–242. doi:10.21136/MB.2022.0128-21

work page doi:10.21136/mb.2022.0128-21 2023

[5] [5]

M. M. Chems-Eddin, M. B. T. El Hamam, and M. A. Hajjami,On the unit group and the2-class number ofQ( √ 2, √p, √q), Ramanujan J.65(2024), 1475–1510. doi:10.1007/s11139-024-00947-x

work page doi:10.1007/s11139-024-00947-x 2024

[6] [6]

M. M. Chems-Eddin and H. El Mamry,On the unit group of some multiquadratic fields, Rend. Circ. Mat. Palermo (2)74(2025), Article 171. doi:10.1007/s12215-025-01274-w

work page doi:10.1007/s12215-025-01274-w 2025

[7] [7]

M. B. T. El Hamam,The unit group of some fields of the formQ( √ 2, √p, √q, √ −ℓ), Math. Bohem. 149(2024), 49–55. doi:10.21136/MB.2023.0077-22

work page doi:10.21136/mb.2023.0077-22 2024

[8] [8]

M. B. T. El Hamam,The unit group of some fields of the formQ( √ 2, √pq, √ps)and Q( √ 2, √pq, √ps, √ −ℓ), Period. Math. Hung.91(2025), 88–100. doi:10.1007/s10998-025-00655-z

work page doi:10.1007/s10998-025-00655-z 2025

[9] [9]

M. B. T. El Hamam,Fundamental system of units of some multiquadratic fields of degree8and degree 16, Ricerche Mat.75(2026), 955–966. doi:10.1007/s11587-025-01015-2

work page doi:10.1007/s11587-025-01015-2 2026

[10] [10]

M. B. T. El Hamam,Erratum: Fundamental systems of units of some multiquadratic fields of degree8 and degree16, Ricerche Mat.75(2026), 967–980. doi:10.1007/s11587-025-01039-8

work page doi:10.1007/s11587-025-01039-8 2026

[11] [11]

Kubota, ¨Uber den bizyklischen biquadratischen Zahlk¨ orper, Nagoya Math

T. Kubota, ¨Uber den bizyklischen biquadratischen Zahlk¨ orper, Nagoya Math. J.10(1956), 65–85. doi:10.1017/S0027763000000088

work page doi:10.1017/s0027763000000088 1956

[12] [12]

23 Alain Robert.A course in p-adic analysis

J. Neukirch,Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Springer, 1999. doi:10.1007/978-3-662-03983-0

work page doi:10.1007/978-3-662-03983-0 1999

[13] [13]

https://doi.org/10.1007/978-1- 4684-3384-5_11

J.-P. Serre,Local Fields, Graduate Texts in Mathematics, vol. 67, Springer, 1979. doi:10.1007/978-1- 4757-5673-9

work page doi:10.1007/978-1- 1979

[14] [14]

Varmon, ¨Uber Abelsche K¨ orper, deren alle Gruppeninvarianten aus einer Primzahl bestehen, und ¨ uber Abelsche K¨ orper als Kreisk¨ orper, Lund, H˚ akan Ohlssons Boktryckeri, 1925

J. Varmon, ¨Uber Abelsche K¨ orper, deren alle Gruppeninvarianten aus einer Primzahl bestehen, und ¨ uber Abelsche K¨ orper als Kreisk¨ orper, Lund, H˚ akan Ohlssons Boktryckeri, 1925

work page 1925

[15] [15]

Wada,On the class number and the unit group of certain algebraic number fields, J

H. Wada,On the class number and the unit group of certain algebraic number fields, J. Fac. Sci. Univ. Tokyo Sect. I13(1966), 201–209. Department of Mathematics, FPT University, Quy Nhon AI Campus, An Phu Thinh New Urban Area, Vietnam Email address:dangphuc150488@gmail.com

work page 1966