Twisted Derivations in Algebraic Number Fields

[2412.03507] Twisted Derivations in Algebraic Number Fields Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2412.03507 (math) [Submitted on 4 Dec 2024 ( v1 ), last revised 9 Apr 2026 (this version, v2)] Title: Twisted Derivations in Algebraic Number Fields Authors: Praveen Manju , Rajendra Kumar Sharma View a PDF of the paper titled Twisted Derivations in Algebraic Number Fields, by Praveen Manju and Rajendra Kumar Sharma View PDF HTML (experimental) 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. Subjects: Number Theory (math.NT) ; Commutative Algebra (math.AC); Rings and Algebras (math.RA) MSC classes: 13N15, 11R04, 94B05, 11R18, 11Y40, 11C20, 12F05, 13B99 Cite as: arXiv:2412.03507 [math.NT] (or arXiv:2412.03507v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2412.03507 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Praveen Manju [ view email ] [v1] Wed, 4 Dec 2024 17:48:59 UTC (21 KB) [v2] Thu, 9 Apr 2026 22:03:42 UTC (26 KB) Full-text links: Access Paper: View a PDF of the paper titled Twisted Derivations in Algebraic Number Fields, by Praveen Manju and Rajendra Kumar Sharma View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2024-12 Change to browse by: math math.AC math.RA References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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