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arxiv: 2412.03507 · v2 · submitted 2024-12-04 · 🧮 math.NT · math.AC· math.RA

Twisted Derivations in Algebraic Number Fields

Praveen Manju , Rajendra Kumar Sharma This is my paper

Pith reviewed 2026-05-23 07:39 UTC · model grok-4.3

classification 🧮 math.NT math.ACmath.RA
keywords twisted derivationsalgebraic number fieldscyclotomic fieldsinner derivationsouter derivationsintegral extensionsHom-IDD codes
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The pith

All A-linear (σ, τ)-derivations from an integral extension B to its splitting field E are classified by their action on the generator θ.

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

The paper classifies every A-linear map D from B = A[θ] to the splitting field E that satisfies the (σ, τ)-derivation rule for ring homomorphisms σ and τ that fix A. A reader cares because the classification extends immediately to algebraic number fields and their rings of integers, giving an explicit way to decide when a derivation is inner. The work supplies conjectures on the precise condition for inner derivations in cyclotomic integer rings for two families of n, uses those conjectures to settle the existence of outer derivations, and applies the classification to build binary Hom-IDD codes.

Core claim

We classify all A-linear maps D: B → E which are (σ, τ)-derivations. Consequently, we classify all (σ, τ)-derivations in certain field extensions, algebraic number fields, and their ring of algebraic integers. For the ring of algebraic integers O_K = Z[ζ] of the cyclotomic number field K = Q(ζ), and a pair (σ, τ) of two different Z-algebra endomorphisms of O_K, we conjecture a necessary and sufficient condition for a (σ, τ)-derivation D:O_K → O_K to be inner. This is done for two different forms of n: (i) n = 2^r p and (ii) n = p^k. As an application we also conjecture the existence and non-existence of non-zero outer derivations of O_K, thus answering the twisted derivation problem in O_K.

What carries the argument

(σ, τ)-derivation: an A-linear map D: B → E obeying the twisted Leibniz identity D(ab) = σ(a)D(b) + D(a)τ(b)

If this is right

  • All (σ, τ)-derivations on the described integral extensions and field extensions are given by an explicit formula involving the image of θ.
  • For O_K in cyclotomic fields with n of the stated forms, a (σ, τ)-derivation is inner precisely when it meets the conjectured relation.
  • Non-zero outer (σ, τ)-derivations exist for one of the two families of n and fail to exist for the other.
  • The classification directly produces new binary Hom-IDD codes.

Where Pith is reading between the lines

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

  • The same classification technique could be run on non-cyclotomic extensions to test whether outer derivations exist there.
  • Computational verification of the conjectures for additional small n would either strengthen or refute the pattern before a general proof is attempted.
  • The link to Hom-IDD codes suggests that automorphism data from number fields may systematically generate codes with prescribed minimum distance.

Load-bearing premise

B equals A adjoined with a single element θ forming an integral extension that is an integral domain whose quotient field has minimal splitting field E over which the two distinct homomorphisms σ and τ are defined and fix A elementwise.

What would settle it

A concrete (σ, τ)-derivation on Z[ζ] for n=4 or n=9 whose value on ζ violates the conjectured algebraic condition yet still satisfies the twisted derivation rule.

read the original abstract

Let $A$ be a commutative ring with unity and $B = A[\theta]$ be an integral extension of $A$. Assume that $B$ is an integral domain with quotient field $\mathbb{K}$ and $\mathbb{E}$ is the minimal splitting field of $\theta$ over $\mathbb{K}$. Suppose $\sigma, \tau: B \rightarrow \mathbb{E}$ are two different ring homomorphisms that fix $A$ element-wise. In this article, we classify all $A$-linear maps $D: B \rightarrow \mathbb{E}$ which are $(\sigma, \tau)$-derivations. Consequently, we classify all $(\sigma, \tau)$-derivations in certain field extensions, algebraic number fields, and their ring of algebraic integers. For the ring of algebraic integers, $O_{\mathbb{K}} = \mathbb{Z}[\zeta]$ of the cyclotomic number field $\mathbb{K} = \mathbb{Q}(\zeta)$ ($\zeta$ an $n^{\text{th}}$ primitive root of unity), and a pair $(\sigma, \tau)$ of two different $\mathbb{Z}$-algebra endomorphisms of $O_{\mathbb{K}}$, we conjecture (using SageMath) a necessary and sufficient condition for a $(\sigma, \tau)$-derivation $D:O_{\mathbb{K}} \rightarrow O_{\mathbb{K}}$ to be inner. This is done for two different forms of $n$: (i) $n = 2^{r}p$ ($r \in \mathbb{N}$ and $p$ an odd rational prime), and (ii) $n=p^{k}$ ($k \in \mathbb{N} \setminus \{1\}$ and $p$ any rational prime). As an application of our main result on classification of $(\sigma, \tau)$-derivations $D:B \rightarrow \mathbb{E}$ and also the conjectures on inner $(\sigma, \tau)$-derivations of $O_{\mathbb{K}}$, we also conjecture the existence and non-existence of non-zero outer derivations of $O_{\mathbb{K}}$ for the above two forms of $n$, thus answering the twisted derivation problem in $O_{\mathbb{K}}$. Finally, as another application of our main result on the classification of $(\sigma, \tau)$-derivations $D:B \rightarrow \mathbb{E}$, we construct some binary Hom-IDD codes in coding theory.

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.

Referee Report

2 major / 4 minor

Summary. The manuscript claims to classify all A-linear (σ, τ)-derivations D: B → E where B = A[θ] is a monogenic integral extension of a commutative unital ring A (B an integral domain with quotient field K, E the minimal splitting field of θ over K, and σ, τ distinct A-fixing homomorphisms B → E). It extends the classification to certain field extensions, algebraic number fields, and rings of algebraic integers. For cyclotomic fields K = Q(ζ_n) with O_K = Z[ζ_n], it conjectures (via SageMath) a necessary and sufficient condition for D: O_K → O_K to be inner when n is of the form 2^r p or p^k; it further conjectures existence/non-existence of nonzero outer derivations on O_K and constructs binary Hom-IDD codes as an application.

Significance. If the central classification is established by explicitly solving the linear conditions imposed by the twisted Leibniz rule and the minimal polynomial of θ, the result supplies a concrete, finite-dimensional parametrization of twisted derivations on monogenic algebras. This is a useful contribution to the study of derivations in algebraic extensions. The conjectures on inner/outer derivations in cyclotomic rings, if substantiated, would address the twisted derivation problem in O_K for the indicated forms of n. The coding-theory application demonstrates one concrete use of the classification.

major comments (2)
  1. [Cyclotomic conjectures] Cyclotomic conjectures paragraph: the necessary and sufficient condition for a (σ, τ)-derivation D: O_K → O_K to be inner is stated to have been obtained computationally but is never written explicitly, nor are the tested values of n, the precise verification procedure, or the edge cases described. This renders the conjecture impossible to check or reproduce from the manuscript alone and weakens its use in the subsequent claim of answering the twisted derivation problem.
  2. [Application to outer derivations] Application to outer derivations: the conjectures on existence and non-existence of nonzero outer derivations on O_K are presented as resolving the twisted derivation problem, yet they rest directly on the unstated inner-condition conjecture; without an explicit statement or independent verification of that condition, the outer-existence claim remains unsupported.
minor comments (4)
  1. The abstract is lengthy and repeats the classification claim several times; a shorter version focused on the main theorem and the precise scope of the conjectures would improve readability.
  2. Notation: the minimal splitting field E is introduced without clarifying its Galois-theoretic relation to the embeddings σ and τ or to the normal closure of K(θ).
  3. The manuscript should supply at least one fully worked low-degree example (e.g., quadratic or cubic θ) that exhibits the explicit form of all (σ, τ)-derivations obtained from the linear system.
  4. References to prior literature on twisted derivations, inner derivations in number fields, and Hom-IDD codes are missing or insufficient.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Cyclotomic conjectures] Cyclotomic conjectures paragraph: the necessary and sufficient condition for a (σ, τ)-derivation D: O_K → O_K to be inner is stated to have been obtained computationally but is never written explicitly, nor are the tested values of n, the precise verification procedure, or the edge cases described. This renders the conjecture impossible to check or reproduce from the manuscript alone and weakens its use in the subsequent claim of answering the twisted derivation problem.

    Authors: We agree that the explicit statement of the conjectured necessary and sufficient condition is needed for reproducibility. In the revised manuscript we will state the condition in full for both families (n=2^r p and n=p^k), list all tested n, describe the SageMath procedure used to derive and verify it, and note the edge cases examined. This directly addresses the concern. revision: yes

  2. Referee: [Application to outer derivations] Application to outer derivations: the conjectures on existence and non-existence of nonzero outer derivations on O_K are presented as resolving the twisted derivation problem, yet they rest directly on the unstated inner-condition conjecture; without an explicit statement or independent verification of that condition, the outer-existence claim remains unsupported.

    Authors: We acknowledge the logical dependence. Once the inner-condition conjecture is stated explicitly (as promised in the response to the first comment), the outer-derivation conjectures become supported by that statement. The revised text will make this dependence explicit and will not claim resolution of the twisted derivation problem beyond what the stated conjectures allow. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit linear classification from definition

full rationale

The central classification of A-linear (σ, τ)-derivations D: B → E for monogenic B = A[θ] proceeds by imposing the twisted Leibniz rule on the power basis of θ together with the minimal polynomial relation, yielding a finite system of linear equations over E whose solutions parametrize all such maps. This is a direct algebraic consequence of the given hypotheses (B integral domain, E minimal splitting field, σ ≠ τ fixing A) and does not reduce to any fitted parameter, self-definition, or prior self-citation. The cyclotomic conjectures are separately labeled as SageMath computations on the inner/outer distinction inside O_K and do not support the general classification. No load-bearing self-citations or ansatzes appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard commutative-algebra assumptions stated in the abstract; no free parameters, invented entities, or non-standard axioms are introduced.

axioms (2)
  • domain assumption B is an integral domain with quotient field K
    Required for the definition of E and the maps D, σ, τ.
  • domain assumption σ, τ are distinct A-fixing ring homomorphisms
    Core to the (σ, τ)-derivation identity.

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Reference graph

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