arxiv: 2604.15928 · v1 · submitted 2026-04-17 · 🧮 math.KT · math.NT

Recognition: unknown

Sums of two symbols in K₂(F)/2K₂(F) in characteristic two

Adam Chapman, Ahmed Laghribi, Demba Barry

Pith reviewed 2026-05-10 07:52 UTC · model grok-4.3

classification 🧮 math.KT math.NT

keywords Milnor K-theorycharacteristic twosymbolschain lemmaK_2 mod 2K_4 mod 2

0 comments

The pith

A sum of two symbols in K_2(F)/2K_2(F) over a field of characteristic two carries a well-defined invariant in K_4(F)/2K_4(F) that vanishes precisely when the sum is congruent to a single symbol in K_2(F)/4K_2(F).

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

The paper first establishes a chain lemma: when two sums of two symbols are congruent in the mod-2 quotient of K_2(F), they can be joined by a finite sequence of elementary replacements of individual symbols that keep the class fixed. This lemma is then used to show that the product of the four entries defines an element of K_4(F)/2K_4(F) that depends only on the congruence class of the original sum. The same lemma yields the characterization that this four-fold product is zero if and only if the two-symbol sum can be rewritten as a single symbol in the quotient by 4K_2(F). The work also supplies upper bounds on the minimal number of symbols needed to represent certain elements in higher 2-power quotients of K_2(F).

Core claim

We prove a chain lemma that connects any two congruent sums A = {a,b}_2 + {c,d}_2 and B = {α,β}_2 + {γ,δ}_2 in K_2(F)/2K_2(F) by a finite sequence of small steps when char(F) = 2. This makes the four-fold symbol {a,b,c,d}_2 a well-defined invariant in K_4(F)/2K_4(F) for the class of A, and shows that the invariant is trivial if and only if A is congruent to a single symbol in K_2(F)/4K_2(F). We also bound the symbol length of an element C in K_2(F)/2^m K_2(F) from above whenever C arises as the sum of at most four symbols in K_2(F)/2^{m+1} K_2(F).

What carries the argument

The chain lemma: any two congruent sums of two symbols in K_2(F)/2K_2(F) can be joined by a finite sequence of elementary replacements of symbols that preserve the congruence class.

If this is right

The four-fold symbol is independent of the choice of representatives for the two-symbol sum.
The invariant vanishes exactly when the sum reduces to length one in the quotient by 4K_2(F).
Elements arising as sums of at most four symbols in K_2(F)/2^{m+1}K_2(F) have symbol length at most a fixed bound when viewed in K_2(F)/2^m K_2(F).

Where Pith is reading between the lines

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

The chain lemma supplies a computational method for deciding whether a given two-symbol sum reduces to length one modulo 4 by checking whether the associated four-fold product is zero.
The invariant distinguishes congruence classes inside K_2(F)/2K_2(F) that survive to the quotient by 4K_2(F).

Load-bearing premise

Any two congruent sums of two symbols in K_2(F)/2K_2(F) can be connected by a finite sequence of small steps consisting of explicit replacements of symbols that preserve the class.

What would settle it

An explicit field F of characteristic 2 together with two congruent sums A and B whose associated four-fold symbols differ in K_4(F)/2K_4(F), or a sum A that cannot be written as a single symbol modulo 4K_2(F) yet has vanishing four-fold invariant.

read the original abstract

In this paper, study sums $A=\{a,b\}_2+\{c,d\}_2$ of two symbols in $K_2(F)/2K_2(F)$ when $\operatorname{char}(F)=2$. We first prove a chain lemma that connects $A$ to $B=\{\alpha,\beta\}_2+\{\gamma,\delta\}_2$ by a finite sequence of small steps when $A \equiv B$. We use this lemma to prove that $\{a,b,c,d\}_2 \in K_4(F)/2K_4(F)$ is a well-defined invariant of $A$, and that this invariant is trivial if and only if $A$ is congruent to a single symbol in $K_2(F)/4K_2(F)$. We also bound the symbol length of $C$ in $K_2(F)/2^m K_2(F)$ from above when $C$ is the sum of up to four symbols in $K_2(F)/2^{m+1}K_2(F)$.

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 chain lemma for connecting congruent two-symbol sums in char-2 K2/2 is the real new piece, and it lets them define a K4/2 invariant that detects single-symbol cases mod 4.

read the letter

The paper proves a chain lemma that links any two congruent sums of two symbols in K2(F)/2K2(F) over a field F of characteristic 2 by a sequence of small replacements. They then use this to show that the four-symbol {a,b,c,d} in K4(F)/2K4(F) is well-defined on the class of the sum and vanishes exactly when the sum is congruent to a single symbol in K2(F)/4K2(F). They also give upper bounds on symbol length for sums of up to four symbols in the higher 2-power quotients K2/2^m K2.

Referee Report

2 major / 2 minor

Summary. The paper studies sums A = {a,b}_2 + {c,d}_2 of two symbols in K_2(F)/2K_2(F) for fields F of characteristic 2. It proves a chain lemma connecting any two equivalent such sums A ≡ B via a finite sequence of small symbol replacements. This lemma is used to establish that the 4-symbol {a,b,c,d}_2 defines a well-defined invariant in K_4(F)/2K_4(F) for the class of A, which vanishes if and only if A is congruent to a single symbol in K_2(F)/4K_2(F). The paper also gives upper bounds on the symbol length of elements in K_2(F)/2^m K_2(F) that arise as sums of at most four symbols from K_2(F)/2^{m+1}K_2(F).

Significance. If the chain lemma holds, the construction supplies a concrete invariant linking 2-torsion in K_2 to K_4, which may help classify elements and relations in Milnor K-theory over fields of characteristic 2. The symbol-length bounds are of independent computational value. The work is technically focused and could be useful for researchers studying quadratic forms, Galois cohomology, or explicit presentations of K-groups in positive characteristic.

major comments (2)

The chain lemma (whose proof occupies the central technical section) is load-bearing for the well-definedness claim. Each elementary replacement step must be shown to preserve the class of {a,b,c,d}_2 in K_4(F)/2K_4(F); the manuscript should explicitly verify this for the characteristic-2 adaptations of the Steinberg relations and any other moves employed. Without a line-by-line check that no step alters the 4-symbol class, the invariance statement remains conditional.
The vanishing criterion (triviality of the 4-symbol invariant iff A is congruent to a single symbol modulo 4K_2(F)) relies on the chain lemma plus an additional reduction argument. The manuscript should isolate the precise place where the single-symbol case is handled and confirm that the argument does not inadvertently assume the result it is proving.

minor comments (2)

Notation for the 4-symbol {a,b,c,d}_2 should be introduced with an explicit reference to the standard definition in K_4(F) (e.g., via the usual symbol map or the presentation of Milnor K-theory) to avoid ambiguity.
The statement of the symbol-length bound would benefit from an explicit example or small-field computation illustrating the improvement over the naive bound of 4.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the chain lemma and vanishing criterion. We address each major comment below and will revise the manuscript accordingly to make the arguments fully explicit.

read point-by-point responses

Referee: The chain lemma (whose proof occupies the central technical section) is load-bearing for the well-definedness claim. Each elementary replacement step must be shown to preserve the class of {a,b,c,d}_2 in K_4(F)/2K_4(F); the manuscript should explicitly verify this for the characteristic-2 adaptations of the Steinberg relations and any other moves employed. Without a line-by-line check that no step alters the 4-symbol class, the invariance statement remains conditional.

Authors: We agree that explicit verification is needed for clarity. The chain lemma in Section 3 is proved by showing each replacement (including the characteristic-2 Steinberg relation and bilinearity moves) preserves the class in K_2(F)/2K_2(F). In the revision we will insert a new Lemma 3.4 immediately after the chain lemma statement that lists every elementary move and directly computes that the associated 4-symbol remains unchanged in K_4(F)/2K_4(F). This makes the well-definedness unconditional without altering the existing proof. revision: yes
Referee: The vanishing criterion (triviality of the 4-symbol invariant iff A is congruent to a single symbol modulo 4K_2(F)) relies on the chain lemma plus an additional reduction argument. The manuscript should isolate the precise place where the single-symbol case is handled and confirm that the argument does not inadvertently assume the result it is proving.

Authors: The single-symbol case is treated by direct computation in the proof of Theorem 4.2 after the chain lemma has reduced the general sum to a single symbol. This computation uses only the definition of symbols in K_4 and the congruence relation in K_2/4K_2; it does not invoke the chain lemma or the general invariance result. To eliminate any appearance of circularity we will extract the single-symbol vanishing as a standalone Lemma 4.1 and then apply the chain lemma in the proof of the general criterion. The logical order will be stated explicitly at the beginning of Section 4. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper first proves the chain lemma connecting equivalent sums A ≡ B via explicit small steps (Steinberg relations and char-2 adaptations), then applies the lemma to establish that {a,b,c,d}_2 is well-defined on the class and trivial precisely when A is a single symbol mod 4K_2(F). This ordering and structure make the derivation self-contained; the well-definedness is a direct consequence of the independently established lemma rather than a reduction to fitted inputs, self-definitions, or load-bearing self-citations. No quoted step equates the target invariant to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only view yields no explicit free parameters or invented entities; the work rests on standard background facts about Milnor K-groups.

axioms (1)

standard math Standard properties and relations of Milnor K-groups K_n(F) in characteristic 2, including the usual Steinberg relations and 2-torsion behavior.
The chain lemma and invariant construction presuppose these known facts about K2 and K4.

pith-pipeline@v0.9.0 · 5501 in / 1328 out tokens · 48962 ms · 2026-05-10T07:52:40.095731+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages

[1]

Albert.Structure of Algebras, volume 24 ofColloquium Publications

A. Albert.Structure of Algebras, volume 24 ofColloquium Publications. American Math. Soc., 1968

1968
[2]

Aravire, B

R. Aravire, B. Jacob, and M. O’Ryan. The de Rham Witt complex, cohomological kernels andp m-extensions in characteristicp.J. Pure Appl. Algebra, 222(12):3891–3945, 2018

2018
[3]

A. Chapman. Symbol length of classes in MilnorK-groups.St. Petersburg Math. J., 34(4):715–720, 2023. Translated from Algebra i Analiz34(2022), No. 4

2023
[4]

A. Chapman. The Cyclicity of Tensor Products of Cyclicp-Algebras. Preprint, arXiv:2510.18421 [math.RA] (2025), 2025

work page arXiv 2025
[5]

Chapman and A

A. Chapman and A. Laghribi. Five-dimensional minimal quadratic and bilinear forms over function fields of conics.Pac. J. Math., 339(2):243–264, 2025

2025
[6]

Chapman, A

A. Chapman, A. Laghribi, and D. Mukhija. Forms inI 3 qFand their symbol length inH 3 2(F). Manuscr. Math., 176(5):22, 2025. Id/No 67

2025
[7]

Elman, N

R. Elman, N. Karpenko, and A. Merkurjev.The algebraic and geometric theory of quadratic forms, volume 56 ofAmerican Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2008

2008
[8]

B. Kahn. Comparison of some field invariants.J. Algebra, 232(2):485–492, 2000. 7

2000
[9]

K. Kato. Symmetric bilinear forms, quadratic forms and MilnorK-theory in characteristic two.Invent. Math., 66(3):493–510, 1982

1982
[10]

Mammone and A

P. Mammone and A. Merkurjev. On the corestriction ofp n-symbol.Isr. J. Math., 76(1-2):73– 80, 1991

1991
[11]

A. S. Sivatski. The chain lemma for biquaternion algebras.J. Algebra, 350:170–173, 2012

2012
[12]

A. S. Sivatski. Degree four cohomological invariants for certain central simple algebras.J. Algebra, 527:337–347, 2019

2019
[13]

A. S. Sivatski. Biquaternion division algebras and fourth power-central elements.J. Pure Appl. Algebra, 226(5):8, 2022. Id/No 106947

2022
[14]

V oevodsky

V . V oevodsky. Motivic cohomology withZ/2-coefficients.Publ. Math., Inst. Hautes Étud. Sci., 98:59–104, 2003. 8

2003