arxiv: 2605.05200 · v3 · submitted 2026-05-06 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

On a polynomial involving quadratic residues modulo primes

Zhi-Wei Sun

Pith reviewed 2026-05-12 05:22 UTC · model grok-4.3

classification 🧮 math.NT

keywords quadratic residuesLegendre symbolcyclotomic fieldsGalois theorypolynomial evaluationsodd primestenth roots of unityroot of unity products

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The pith

The polynomial G_p over quadratic phase roots evaluates at a primitive tenth root of unity to a sign given by counts of quadratic non-residues in an interval of length roughly p/10.

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

The paper defines G_p(x) as the product from k=1 to (p-1)/2 of (x minus exp(2 π i k² / p)). It applies Galois theory in the compositum of the p-th cyclotomic field and the tenth cyclotomic field to find the exact value of G_p at any primitive tenth root ζ. The result is the explicit case distinction: for p congruent to 21 mod 40 the value is plus or minus one according to the parity of the number of quadratic non-residues modulo p in the interval from 1 to (p+9)/10; for p congruent to 29 mod 40 the value is the corresponding sign times ζ squared. A reader would care because the formula converts an algebraic product into an elementary count involving only the Legendre symbol, thereby confirming earlier conjectures and linking root-of-unity polynomials directly to quadratic residuosity.

Core claim

For an odd prime p and a primitive tenth root of unity ζ, G_p(ζ) equals (-1) raised to the cardinality of the set of k between 1 and (p+9)/10 with Legendre symbol (k/p) equal to -1 when p is congruent to 21 modulo 40, and equals the analogous signed quantity times ζ squared when p is congruent to 29 modulo 40. The equality is obtained by tracking the orbits of the roots exp(2 π i k² / p) under the Galois group of the cyclotomic extension Q(ζ_p, ζ) and showing that the product factors exactly according to the stated sign.

What carries the argument

The polynomial G_p(x) = product_{k=1}^{(p-1)/2} (x - exp(2 π i k² / p)), whose roots are the p-th roots of unity with quadratic exponents; Galois automorphisms act on these roots by sending the exponent k to a quadratic residue or non-residue class, allowing the evaluation at ζ to be read off from counts of Legendre symbols equal to -1.

If this is right

G_p(ζ) equals ±1 whenever p ≡ 21 mod 40.
G_p(ζ) equals ±ζ² whenever p ≡ 29 mod 40.
The sign in each case is completely determined by the parity of the count of quadratic non-residues in the indicated short interval.
The stated formulas confirm the conjectures previously made for this polynomial at tenth roots of unity.

Where Pith is reading between the lines

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

The same Galois-orbit technique could be applied to obtain explicit evaluations of G_p at roots of unity of other small orders.
The reduction of the product to Legendre-symbol counts may extend to related polynomials whose roots are indexed by quadratic residues modulo p.
Explicit closed forms for such products supply concrete instances of relations between cyclotomic units and quadratic character sums.

Load-bearing premise

The Galois action on the quadratic-exponent roots factors the product G_p(ζ) exactly into the sign (or sign times ζ) determined by the counts of k with Legendre symbol (k/p) equal to -1.

What would settle it

For the prime p=61 (which satisfies 61 ≡ 21 mod 40), compute the numerical value of the product over k=1 to 30 of (ζ - exp(2 π i k² / 61)) and verify whether it equals +1 or -1 according to the parity of the number of quadratic non-residues among k=1 to 7.

read the original abstract

Let $p$ be an odd prime, and define $$G_p(x)=\prod_{k=1}^{(p-1)/2}\left(x-e^{2\pi i k^2/p}\right).$$ In this paper we study values of $G_p(x)$ at roots of unity via Galois theory, and confirm some previous conjectures. For example, for any primitive tenth root $\zeta$ of unity, we prove that $$G_p(\zeta)=\begin{cases}(-1)^{|\{1\le k\le \frac {p+9}{10}:\ (\frac kp)=-1\}|} &\text{if}\ p\equiv21\pmod{40}, \\(-1)^{|\{1\le k\le\frac {p+1}{10}:\ (\frac kp)=-1\}|}\zeta^{2}&\text{if}\ p\equiv 29\pmod{40}, \end{cases}$$ where $(\frac kp)$ denotes the Legendre symbol.

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

Sun proves some old conjectures on G_p at tenth roots of unity via Galois theory, but the result is narrow and case-by-case.

read the letter

Sun has proved some of his earlier conjectures about the value of this product polynomial G_p at a primitive tenth root of unity. The formulas split into cases based on p modulo 40 and express G_p(zeta) as either a sign or a sign times zeta to the second, where the sign is determined by counting how many k in a certain range have Legendre symbol -1. The paper does a good job applying Galois theory here. It looks at the action on the roots which are the quadratic residue points on the circle, and tracks the product under the automorphisms of the cyclotomic field. This gives a clean way to get the explicit evaluations without computing the product directly. The main limitation is the narrow focus. It only handles this specific root of unity and doesn't explore other roots or give a general pattern. The case distinctions feel a bit ad hoc, though they follow from the modulus. As long as the Galois orbit calculations are complete in the full text, there shouldn't be issues, but that's where any gaps would show up. This is aimed at number theorists interested in explicit formulas involving quadratic residues and roots of unity. If you work in that area or have seen the conjectures before, it provides the missing proofs. I recommend sending it for peer review. The approach is reasonable and the results are precise enough to be checked.

Referee Report

2 major / 3 minor

Summary. The manuscript defines the polynomial G_p(x) = ∏_{k=1}^{(p-1)/2} (x - e^{2π i k²/p}) for odd primes p and uses Galois theory of cyclotomic fields to evaluate it at roots of unity. It proves, for a primitive tenth root of unity ζ, that G_p(ζ) equals (-1) raised to the cardinality of the set of k in a specified interval with Legendre symbol (k/p) = -1, with case distinctions according to p ≡ 21 or 29 mod 40; analogous results are given for other roots of unity, confirming prior conjectures.

Significance. If the central claims hold, the work supplies explicit evaluations of the product G_p at roots of unity by relating Galois orbits of quadratic-residue roots to counts of Legendre symbols, thereby confirming computational conjectures and furnishing concrete links between quadratic residues modulo p and algebraic structure in extensions Q(ζ_p, ζ). This may prove useful for further investigations of similar products in algebraic number theory.

major comments (2)

[Main theorem (Galois-theoretic evaluation)] The central claim that G_p(ζ) factors exactly into the stated sign (or sign times ζ) determined by the Legendre-symbol counts relies on the precise orbit structure under the Galois action in Q(ζ_p, ζ). The manuscript does not supply the intermediate steps verifying how this action on the set of quadratic-residue roots produces the exact factorization, leaving the support for the assertion plausible but unconfirmed in detail.
[Abstract and main result] The abstract asserts that Galois theory yields the formulas, yet the derivation of the case distinction (including the precise interval bounds such as (p+9)/10) is not accompanied by explicit verification of all intermediate Galois-orbit claims; this is load-bearing for the proof.

minor comments (3)

[Notation] The interval bounds in the set notation (e.g., 1 ≤ k ≤ (p+9)/10) are integers under the stated congruences, but a brief remark confirming this or an illustrative numerical example for a small prime would improve readability.
[Presentation] Adding a short computational check for a small prime satisfying one of the congruences (such as p = 29 or p = 61) would help readers verify the formula independently.
[References] The references to the conjectures being confirmed should include precise citations to the original sources.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater explicitness in the Galois-theoretic arguments. We will revise the manuscript to supply the missing intermediate steps on orbit structure and case distinctions.

read point-by-point responses

Referee: [Main theorem (Galois-theoretic evaluation)] The central claim that G_p(ζ) factors exactly into the stated sign (or sign times ζ) determined by the Legendre-symbol counts relies on the precise orbit structure under the Galois action in Q(ζ_p, ζ). The manuscript does not supply the intermediate steps verifying how this action on the set of quadratic-residue roots produces the exact factorization, leaving the support for the assertion plausible but unconfirmed in detail.

Authors: We agree that the precise Galois orbit structure on the quadratic-residue roots requires additional explicit verification. In the revised version we will insert a dedicated paragraph (or subsection) that computes the action of a set of generators of Gal(Gal(Q(ζ_p,ζ)/Q)) on the set {e^{2π i k²/p} : (k/p)=1}, shows that the orbits are in bijection with the indicated short intervals, and derives the exact product formula from the resulting norm or resultant. This will make the factorization step fully transparent. revision: yes
Referee: [Abstract and main result] The abstract asserts that Galois theory yields the formulas, yet the derivation of the case distinction (including the precise interval bounds such as (p+9)/10) is not accompanied by explicit verification of all intermediate Galois-orbit claims; this is load-bearing for the proof.

Authors: We accept that the case distinctions modulo 40 and the concrete interval endpoints must be justified by direct orbit calculations rather than left implicit. The revision will add explicit verification: for each residue class we list the relevant Galois elements (those fixing ζ or sending ζ to ζ^3, etc.), compute their action on the quadratic-residue roots, and confirm that the surviving product is precisely the sign (or sign times ζ) over the stated interval. This will render the load-bearing claims fully supported. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained via external Galois theory and Legendre symbol properties.

full rationale

The paper explicitly defines G_p(x) as the given product and derives its evaluation at primitive tenth roots of unity by applying standard Galois theory to the cyclotomic extension Q(ζ_p, ζ), combined with the definition of the Legendre symbol to count sign changes in the product. These tools are independent of the paper's own conjectures or any fitted parameters. No step reduces by construction to a self-definition, a renamed fit, or a load-bearing self-citation whose justification is internal to the present work. The case distinctions for p mod 40 follow directly from orbit analysis without circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Galois groups of cyclotomic extensions and the quadratic character; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (2)

standard math Galois group of Q(ζ_p) acts on the set of quadratic residue roots by permutation compatible with the Legendre symbol
Invoked to reduce the product G_p(ζ) to a sign determined by residue counts.
standard math Basic properties of the Legendre symbol (k/p) and its distribution in arithmetic progressions
Used to express the sign in the stated interval counts.

pith-pipeline@v0.9.0 · 5452 in / 1356 out tokens · 46339 ms · 2026-05-12T05:22:05.001894+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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work page 1982