REVIEW 3 major objections 4 minor 7 references
Squares in $\mathbb{F}_{p^2}$ and permutations involving primitive roots
T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The sign of the permutation that orders the nonzero squares of $\mathbb{F}_{p^2}$ by powers of a primitive root is completely determined by $p$, a parity bit $\beta_0$, and the class number $h(-p)$ when $p\equiv 3\pmod 4$.
desk verdict A correct and genuinely new sign formula with a repairable gap in Lemma 2.3 that should be fixed before publication. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the difference-product formula for the sign of a permutation of a finite set: if $\tau$ permutes $\{\alpha_1,\ldots,\alpha_n\}$, then $\operatorname{sgn}(\tau)=\prod_{s<t}(\tau(\alpha_t)-\tau(\alpha_s))/(\alpha_t-\alpha_s)$. Applied to the squares of $\mathbb{F}_{p^2}$, this turns the sign of $\sigma_p(g)$ into a ratio of two products. The numerator $\prod_{s<t}(g^{2t}-g^{2s})$ is evaluated by Lemma 2.5: the cyclotomic polynomial $\Phi_{p^2-1}(x)$ divides $F(x)-T(x)$, where $F(x)=\prod_{s<t}(x^{2t}-x^{2s})$ and $T(x)$ is an explicit monomial with sign factor $(-1)^{(p^2+7)/8}$, so at the primitive root $g$ the numerator becomes a known power of $g$. The denominator is factored as $A_p^{(p-1)(p-3)/4}B_p^{(p-1)/2}D_p^{(p-1)/2}$ times a Vandermonde square, with Lemmas 2.1 through 2.4 giving its residue modulo $p$. The parity bit $\beta_0$, defined in (1.6), records which power of the $(p^2-1)$-st root of unity represents $\sqrt{\Delta}$ and converts the remaining powers of $g$ to $\pm 1$.
What would settle it
For a small prime such as $p=5$ or $p=7$, list all nonzero squares of $\mathbb{F}_{p^2}$ in the order $S$ and in the order $S^*$, count inversions to get $\operatorname{sgn}(\sigma_p(g))$, and compare it with the corresponding case of Theorem 1.1; any mismatch refutes the formula. The more targeted check is to enumerate the four counts in (2.4)-(2.7) for these primes.
Extended reading notes
Core claim
The central discovery is Theorem 1.1: for an odd prime $p=2n+1$, with $\beta_0\in\{0,1\}$ defined by $(-1)^{\beta_0}\equiv (\sqrt{\Delta})^{(p-1)/2}\zeta_{p^2-1}^{(p^2-1)/4}\pmod p$, the sign of the permutation $\sigma_p(g)$ is $(-1)^{\beta_0+(p+3)/4}$ when $p\equiv1\pmod4$, $(-1)^{(h(-p)+1)/2+\beta_0}$ when $p\equiv3\pmod4$ and $p>3$, and $(-1)^{1+\beta_0}$ when $p=3$, where $h(-p)$ is the class number of $\mathbb{Q}(\sqrt{-p})$. The formula is complete: no condition on $\Delta$ or $g$ remains beyond the parity bit $\beta_0$. The proof works by writing the sign as a product of differences, evaluating the numerator through a cyclotomic divisibility relation and the denominator through a factorization into three products $A_p$, $B_p$, $D_p$ whose congruences are computed separately.
Load-bearing premise
The argument depends on the four counts of pairs of quadratic residues stated in (2.4)-(2.7) of Lemma 2.3; if any one of those counts is wrong, the denominator evaluation and hence the sign formula fail.
Editorial extensions
If this is right
- For $p\equiv1\pmod4$, once $\beta_0$ is computed the sign is known immediately, with no class-number input.
- For $p\equiv3\pmod4$, $p>3$, the formula means the parity of $(h(-p)+1)/2$ can be recovered from the sign of this explicit permutation.
- Lemma 2.5 supplies a closed congruence for the Vandermonde-type product $\prod_{s<t}(g^{2t}-g^{2s})$ modulo $p$, which can be reused in other $\mathbb{F}_{p^2}$ permutation problems.
- The dependence on the chosen primitive root $g$ is reduced to the parity bit $\beta_0$; the formula itself is otherwise uniform in $g$.
Reading between the lines
- The same difference-product method should apply to permutations of $m$-th powers in $\mathbb{F}_{p^r}$ when the relevant cyclotomic polynomial admits a product identity like Lemma 2.5; this paper does not pursue that extension.
- The four counts in Lemma 2.3, labelled 'one can easily verify', look like special cases of a general count of representations by binary quadratic forms modulo $p$; if so, that underlying structure could both justify and generalize the argument.
- For fixed small $p$, the parity bit $\beta_0$ can be evaluated directly from (1.6), so the theorem gives an explicit finite algorithm for the sign rather than only an existence statement.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines, for an odd prime p=2n+1, a permutation σ_p(g) of the nonzero squares of F_{p^2}: the domain is ordered as the powers g^2,g^4,...,g^{p^2-1} of a primitive root g, and the codomain is ordered as the sequence of squares a_k^2 j^2 (0≤k≤p−1, 1≤j≤n) followed by 1^2,...,n^2, where a_k=k+√Δ and Δ is a quadratic non-residue with Δ≡3 mod 4. The main theorem (Theorem 1.1) gives a complete sign formula for this permutation in terms of the class number h(−p), the residue of p modulo 4, and an auxiliary bit β_0 defined by the congruence (1.6). The proof proceeds through congruence lemmas for products involving √Δ, a cyclotomic evaluation (Lemma 2.5), and an analysis of the numerator and denominator of the permutation sign.
Significance. If correct, the theorem is a substantial extension of Zolotarev-type sign computations to the quadratic extension F_{p^2}, and it connects the sign to the class number h(−p) and to a congruence parameter β_0 that carries information about the choice of primitive root. The claimed formula is precise and falsifiable, and it generalizes earlier work by Sun and the author. The paper also highlights an interesting use of the local existence theorem to identify Q_p(ζ_{p^2−1}) with Q_p(√Δ). I independently checked the formula for p=3 and p=5, and the main theorem's statement matches direct permutation computations in those cases.
major comments (3)
- [Section 2, Lemma 2.3] In the proof of Lemma 2.3, after observing that ∏_{1≤s≤m}(x+s)∏_{1≤t≤m}(x−t) ≡ x^{p−1}−1, the text claims that ∏_{t}(√Δ−t) ≡ −2∏_{s}(√Δ+s) (mod p). This inference is false; for p=5, Δ=3, m=2, the left side is 2i and the right side is 4i in F_25. The desired conclusion (2.3) can nonetheless be derived directly by raising the product identity to the m-th power, so the lemma statement is true, but the published proof contains an invalid step. This step is load-bearing because (2.3) feeds into Lemma 2.4 and the denominator evaluation (2.14).
- [Section 2, proof of Theorem 1.1, definition of β and β_0] After setting α = (p+1)/2 + (p+1)β, the derivation gives (√Δ)^{(p−1)/2} ≡ ζ^{(p^2−1)/4}(−1)^β, so the correct relation is (−1)^β ≡ (√Δ)^{(p−1)/2} ζ^{−(p^2−1)/4}. The proof instead states (−1)^β ≡ (√Δ)^{(p−1)/2} ζ^{(p^2−1)/4}. Consequently one obtains β ≡ β_0+1 (mod 2), not β ≡ β_0 as asserted. Since the subsequent substitutions in Cases 1 and 2 replace β by β_0, this sign error is load-bearing for the final formula.
- [Section 2, Case 1 of proof of Theorem 1.1] The displayed formula sgn(σ_p) ≡ g^{(p^2−1)/4 + ((p−1)/2)α + ((p−1)^2/4)α} does not follow from (2.12) and (2.14). For p=5, Δ=3, α=9, this expression is g^{60} ≡ g^{12} ≡ −1, but the actual sign of σ_5 is +1 and the theorem's formula gives +1. The correct combination of (2.12) and (2.14) introduces a factor (2/p)(−2/p)^{(p+1)/2} and a different dependence on α; after that correction the final formula can be recovered, so the theorem statement appears correct, but this is a genuine error in the proof. Case 2 contains analogous issues.
minor comments (4)
- [Section 2, Lemma 2.3, identities (2.4)-(2.7)] The counting identities (2.4)-(2.7) are asserted with 'one can easily verify' and are used to evaluate a Legendre-symbol product that contributes to the final sign. They are elementary, but the paper should provide a proof or a reference so that the derivation is self-contained.
- [Section 2, Lemma 2.1 and Lemma 2.4] The phrases 'one may easily get' (Lemma 2.1) and 'one may easily verify' (Lemma 2.4) leave important congruence computations to the reader. Expanding these steps would improve the rigor of the proof.
- [Introduction and formatting] There is a typo in the Introduction ('Proposit on' instead of 'Proposition'), and the sequence S in the abstract is typeset with unclear spacing; these should be cleaned up in revision.
- [References] Reference [6] is listed as 'In press'; if it has appeared or has a more complete citation, that should be updated.
Circularity Check
No circularity: the sign formula is derived from independent lemmas and standard class-number input, not from its conclusion.
full rationale
The derivation chain is self-contained and non-circular. The sign of σ_p(g) is expressed as a quotient; the numerator is evaluated using Lemma 2.5 together with g ≡ ζ (mod p), giving (2.12), and the denominator is factored into the products A_p, B_p, D_p, and a fixed Vandermonde-type product (2.13) from Sun [4]. Lemmas 2.1–2.4 are proved directly from finite-field congruences and the class-number formula for Q(√−p). The quantity β0 defined in (1.6) is a well-defined congruence invariant of the chosen primitive root g and square root of Δ; it is not fitted to the target sign, and the final expression is obtained by ordinary p-adic exponent arithmetic. The author's earlier works [5] and [6] are cited only as background on permutations of quadratic residues, and they are not load-bearing in the proof of Theorem 1.1. The only substantive concern visible in the manuscript is a proof gap in Lemma 2.3: after the product identity, the line ∏_t(√Δ−t) ≡ −2∏_s(√Δ+s) does not follow and is false for p=5; this is a correctness issue in the proof as printed, not a circular reduction of the theorem to its assumptions. Since no equation or fitted parameter is equivalent by construction to the claimed sign formula, the circularity score is 0.
Assumptions & free parameters
assumptions (4)
- standard math Q_p(ζ_{p^2-1}) = Q_p(√Δ) as unramified extensions of Q_p
- standard math Product identity ∏_{1≤k≤(p-1)/2}(x-k^2) ≡ x^{(p-1)/2}-1 (mod p)
- standard math Class number formula for Q(√-p) connecting counts of quadratic non-residues to h(-p)
- domain assumption The counting identities (2.4) to (2.7) for representability of Δ by sums of two squares
Cite this review
Pith. "Pith review of Squares in $\mathbb{F}_{p^2}$ and permutations involving primitive roots." pith.science (2026). https://pith.science/paper/EKDLE2UA
@misc{pith2026190807641,
author = {Pith},
title = {Pith review of: Squares in $\mathbbF_p^2$ and permutations involving primitive roots},
year = {2026},
howpublished = {\url{https://pith.science/paper/EKDLE2UA}},
note = {Machine review of arXiv:1908.07641}
}
abstract
Let $p=2n+1$ be an odd prime, and let $\zeta_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. We let $g\in\mathbb{Z}_p[\zeta_{p^2-1}]$ be a primitive root modulo $p\mathbb{Z}_p[\zeta_{p^2-1}]$ with $g\equiv \zeta_{p^2-1}\pmod {p\mathbb{Z}_p[\zeta_{p^2-1}]}$. Let $\Delta\equiv3\pmod4$ be an arbitrary quadratic non-residue modulo $p$ in $\mathbb{Z}$. By the Local Existence Theorem we know that $\mathbb{Q}_p(\sqrt{\Delta})=\mathbb{Q}_p(\zeta_{p^2-1})$. For all $x\in\mathbb{Z}[\sqrt{\Delta}]$ and $y\in\mathbb{Z}_p[\zeta_{p^2-1}]$ we use $\bar{x}$ and $\bar{y}$ to denote the elements $x\mod p\mathbb{Z}[\sqrt{\Delta}]$ and $y\mod p\mathbb{Z}_p[\zeta_{p^2-1}]$ respectively. If we set $a_k=k+\sqrt{\Delta}$ for $0\le k\le p-1$, then we can view the sequence $$S := \overline{a_0^2}, \cdots, \overline{a_0^2n^2}, \cdots,\overline{a_{p-1}^2}, \cdots, \overline{a_{p-1}^2n^2}\cdots, \overline{1^2}, \cdots,\overline{n^2}$$ as a permutation $\sigma$ of the sequence $$S^* := \overline{g^2}, \overline{g^4}, \cdots,\overline{g^{p^2-1}}.$$ We determine the sign of $\sigma$ completely in this paper.
Reference graph
Works this paper leans on
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[5]
L.-Y Wang and H.-L Wu, Applications of Lerch’s theorem to permutations of qua- dratic residues, Bull. Aust. Math. Soc. 100 (2019), 362–371
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W. Duke and K. Hopkins, Quadratic reciprocity in a finite group , Amer. Math. Monthly 112(2005), 251–256
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Neukirch, Algebraic Number Theory, Springer-Verlag Berlin He idelberg, 1999
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[4]
Z.-W Sun, Quadratic residues and related permutations and identitie s, Finite Fields Appl. 59 (2019), 246–283
work page 2019
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[7]
Zolotarev, Nouvelle d´ eonstration de la loi de r´ eciprocit´ e de Legendre, Nouvelles Ann
G. Zolotarev, Nouvelle d´ eonstration de la loi de r´ eciprocit´ e de Legendre, Nouvelles Ann. Math. 11(1872), 354–362. (Hai-Liang Wu) Department of Mathematics, Nanjing Univers ity, Nan- jing 210093, People’s Republic of China E-mail address : whl.math@smail.nju.edu.cn
Reviewed August 14, 2026 · model on record in the stance chip above.
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