On the reciprocity law in $\mathbb{F}_{q}[t]$

Enci Wang; Su Hu

arxiv: 2604.20892 · v1 · submitted 2026-04-21 · 🧮 math.NT

On the reciprocity law in mathbb{F}_(q)[t]

Su Hu , Enci Wang This is my paper

Pith reviewed 2026-05-10 01:41 UTC · model grok-4.3

classification 🧮 math.NT

keywords reciprocity lawpower residue symbolfunction fieldF_q[t]elementary proofcoset representationChinese Remainder TheoremRousseau's method

0 comments

The pith

Rousseau's coset-comparison method yields an elementary proof of the reciprocity law for dth power residue symbols in F_q[t] when d divides q-1.

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

The paper extends a 1991 group-theoretic proof of quadratic reciprocity to the polynomial ring over a finite field. It shows that comparing two distinct coset representations of the group (F_q[t]/(p)^* × F_q[t]/(q)^*)/U via the Chinese Remainder Theorem produces the reciprocity law for the dth power residue symbol without using Gauss's lemma. A sympathetic reader would see this as evidence that the integer proof's algebraic skeleton transfers directly to function fields, offering a uniform elementary route to power reciprocity in both settings. The result matters because it replaces analytic or class-field machinery with direct counting of cosets in a ring where unique factorization holds.

Core claim

By identifying the group (R_p^* × R_q^*)/U with its image under the Chinese Remainder Theorem and comparing the two natural coset decompositions, one obtains the explicit reciprocity relation between the dth power residue symbols (a/p)_d and (a/q)_d for distinct monic irreducibles p and q in F_q[t], valid whenever d divides q-1.

What carries the argument

The comparison of two coset representations of the quotient group (F_q[t]/(p)^* × F_q[t]/(q)^*)/U induced by the Chinese Remainder Theorem isomorphism.

If this is right

The reciprocity law holds for every divisor d of q-1.
No appeal to Gauss's lemma or analytic continuation is required.
The same group-theoretic counting works uniformly for all such d.
The method stays inside elementary algebra once the residue rings are identified.

Where Pith is reading between the lines

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

The same coset comparison may adapt to other Dedekind domains where a Chinese Remainder Theorem isomorphism exists between residue rings.
One could test whether the argument produces explicit formulas for higher-power symbols in global function fields of higher genus.
The approach suggests that many classical reciprocity proofs relying on Gauss sums might admit purely multiplicative-group versions in characteristic p.

Load-bearing premise

The coset representatives and index calculations that work for Z carry over verbatim to F_q[t] once the Chinese Remainder Theorem is applied to the product of residue rings.

What would settle it

A concrete counter-example would be two distinct monic irreducibles p and q together with an element a such that the computed product of the two coset indices differs from the value predicted by the known reciprocity law for some d dividing q-1.

read the original abstract

In 1991, Rousseau gave a new proof of Gauss's quadratic reciprocity by comparing two distinct coset representations of the group $(\mathbb{Z}_{p}^{*} \times \mathbb{Z}_{q}^{*}) / U$ using the Chinese Remainder Theorem, without Gauss's Lemma. In this paper, we extend Rousseau's approach to $\mathbb{F}_{q}[t]$, providing a new, elementary proof of the reciprocity law for the $d$th power residue symbol, where $d$ is any divisor of $q-1$.

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

This paper adapts Rousseau's coset comparison to give an elementary proof of dth-power reciprocity in F_q[t] for d dividing q-1, and the translation works cleanly.

read the letter

The main takeaway is that the authors extend Rousseau's 1991 coset-representative argument from Z to F_q[t] and obtain a proof of the reciprocity law for the dth power residue symbol whenever d divides q-1. The result itself is known, but the method is new in this setting. They work with the quotient group of units modulo a product of two distinct irreducibles, split it via the Chinese Remainder Theorem into a product of two residue fields, embed the constant units diagonally, and compare two natural sets of coset representatives to read off the symbol. Reduced polynomials of degree less than the irreducible play the role that residues mod p and q played in the integer case. The stress-test confirms the finite-field arithmetic introduces no extra obstructions, so the steps carry over directly. That is the part the paper does well: it stays elementary, avoids Gauss's lemma, and keeps the bookkeeping identical to Rousseau's original write-up. The group isomorphism and the choice of representatives are standard and correctly applied. The soft spots are proportionate and limited. The argument is essentially a faithful translation, so once the setup is written down the comparison follows in the same way as over Z. No new group-theoretic machinery appears, and the paper does not claim any. Readers who already know the integer proof will see the function-field version quickly. There are no circular steps or fitted quantities; everything rests on CRT and the structure of the multiplicative groups of finite fields. This paper is for number theorists who work in function-field arithmetic or who collect elementary proofs of reciprocity laws. Someone looking for a short, self-contained argument in the polynomial ring would find it useful. It is not a major advance, but the execution is solid and the claim is precise. I would send it to peer review. The mathematics checks out on its own terms and the extension is worth recording in the literature.

Referee Report

0 major / 3 minor

Summary. The manuscript extends Rousseau's 1991 coset-comparison proof of quadratic reciprocity to the ring F_q[t]. For distinct monic irreducibles p and q, it applies the Chinese Remainder Theorem to obtain an isomorphism F_q[t]/(pq)^* ≅ R_p^* × R_q^* (with R_p = F_q[t]/(p)), identifies the diagonal subgroup U of constant units F_q^*, and compares two sets of coset representatives (reduced polynomials of degree < deg(p) and < deg(q)) to derive the reciprocity relation for the d-th power residue symbol whenever d divides q-1.

Significance. If the details are carried through correctly, the paper supplies an elementary, group-theoretic proof of the d-th power reciprocity law in F_q[t] that avoids Gauss sums, class-field theory, or explicit evaluation of symbols. It directly transplants the integer-case argument and therefore offers a transparent alternative to existing proofs in the function-field literature.

minor comments (3)

§2: the notation for the d-th power residue symbol (·/·)_d is introduced without an explicit definition or reference to the standard normalization in F_q[t]; a one-line reminder of the definition would help readers.
§3.2, after the statement of the main theorem: the comparison of the two coset representatives is summarized rather than written out in full; expanding the final step that equates the two expressions for the symbol would make the argument self-contained.
The paper assumes throughout that p and q are distinct monic irreducibles; a brief sentence confirming that the result extends to the case where one or both are units or constants would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The summary accurately captures the extension of Rousseau's coset-comparison method via the Chinese Remainder Theorem to obtain an elementary proof of the d-th power reciprocity law in F_q[t]. We will make the minor adjustments needed to ensure all details are presented clearly and correctly.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper directly transplants Rousseau's 1991 coset-comparison argument to F_q[t] via the standard Chinese Remainder Theorem isomorphism (F_q[t]/(pq))^* ≅ R_p^* × R_q^* and the diagonal embedding of F_q^*. Reduced representatives are chosen as polynomials of degree less than deg(p) and deg(q); their comparison yields the d-th power reciprocity when d | (q-1). No equation reduces to a fitted input, no self-citation is load-bearing, and the construction does not presuppose the target law. The argument is independent of the result it proves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the Chinese Remainder Theorem to quotients of F_q[t] and on the multiplicative group structure of units in this ring being sufficiently analogous to the integer case.

axioms (2)

standard math Chinese Remainder Theorem holds for the relevant rings and ideals in F_q[t]
Invoked to equate the two coset representations of the group.
domain assumption The unit group structure and coset decompositions in F_q[t] are directly analogous to those in Z
The extension assumes this analogy permits the same comparison without new obstructions.

pith-pipeline@v0.9.0 · 5373 in / 1263 out tokens · 43370 ms · 2026-05-10T01:41:04.229112+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Rousseau,On the quadratic reciprocity law, J

G. Rousseau,On the quadratic reciprocity law, J. Austral. Math. Soc. Ser. A51 (1991), no. 3, 423–425

work page 1991
[2]

Rosen,Number theory in function fields.Graduate Texts in Mathematics, 210

M. Rosen,Number theory in function fields.Graduate Texts in Mathematics, 210. Springer-Verlag, New York, 2002. Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China Email address:mahusu@scut.edu.cn Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China Email address...

work page 2002

[1] [1]

Rousseau,On the quadratic reciprocity law, J

G. Rousseau,On the quadratic reciprocity law, J. Austral. Math. Soc. Ser. A51 (1991), no. 3, 423–425

work page 1991

[2] [2]

Rosen,Number theory in function fields.Graduate Texts in Mathematics, 210

M. Rosen,Number theory in function fields.Graduate Texts in Mathematics, 210. Springer-Verlag, New York, 2002. Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China Email address:mahusu@scut.edu.cn Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China Email address...

work page 2002