Twisted Derivations in Algebraic Number Fields
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2024
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| author | Manju, Praveen Sharma, Rajendra Kumar |
| author_facet | Manju, Praveen Sharma, Rajendra Kumar |
| contents | Let $A$ be a commutative ring with unity and $B = A[θ]$ 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 $θ$ over $\mathbb{K}$. Suppose $σ, τ: B \rightarrow \mathbb{E}$ 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 $(σ, τ)$-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, $O_{\mathbb{K}} = \mathbb{Z}[ζ]$ of the cyclotomic number field $\mathbb{K} = \mathbb{Q}(ζ)$ ($ζ$ an $n^{\text{th}}$ primitive root of unity), and a pair $(σ, τ)$ of two different $\mathbb{Z}$-algebra endomorphisms of $O_{\mathbb{K}}$, we conjecture (using SageMath) a necessary and sufficient condition for a $(σ, τ)$-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 $(σ, τ)$-derivations $D:B \rightarrow \mathbb{E}$ and also the conjectures on inner $(σ, τ)$-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 $(σ, τ)$-derivations $D:B \rightarrow \mathbb{E}$, we construct some binary Hom-IDD codes in coding theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03507 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twisted Derivations in Algebraic Number Fields Manju, Praveen Sharma, Rajendra Kumar Number Theory Commutative Algebra Rings and Algebras 13N15, 11R04, 94B05, 11R18, 11Y40, 11C20, 12F05, 13B99 Let $A$ be a commutative ring with unity and $B = A[θ]$ 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 $θ$ over $\mathbb{K}$. Suppose $σ, τ: B \rightarrow \mathbb{E}$ 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 $(σ, τ)$-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, $O_{\mathbb{K}} = \mathbb{Z}[ζ]$ of the cyclotomic number field $\mathbb{K} = \mathbb{Q}(ζ)$ ($ζ$ an $n^{\text{th}}$ primitive root of unity), and a pair $(σ, τ)$ of two different $\mathbb{Z}$-algebra endomorphisms of $O_{\mathbb{K}}$, we conjecture (using SageMath) a necessary and sufficient condition for a $(σ, τ)$-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 $(σ, τ)$-derivations $D:B \rightarrow \mathbb{E}$ and also the conjectures on inner $(σ, τ)$-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 $(σ, τ)$-derivations $D:B \rightarrow \mathbb{E}$, we construct some binary Hom-IDD codes in coding theory. |
| title | Twisted Derivations in Algebraic Number Fields |
| topic | Number Theory Commutative Algebra Rings and Algebras 13N15, 11R04, 94B05, 11R18, 11Y40, 11C20, 12F05, 13B99 |
| url | https://arxiv.org/abs/2412.03507 |