2024/12/04 by Praveen Manju, Manju, Praveen, R. K. Sharma +1
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.2412.03507
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 \mathbbK and 𝔼 is the minimal splitting field of θ over \mathbbK. Suppose σ, τ: B → 𝔼 are two different ring homomorphisms that fix A element-wise. In this article, 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_\mathbbK = ℤ[ζ] of the cyclotomic number field \mathbbK = ℚ(ζ) (ζ an nth primitive root of unity), and a pair (σ, τ) of two different ℤ-algebra endomorphisms of O_\mathbbK, we conjecture (using SageMath) a necessary and sufficient condition for a (σ, τ)-derivation D:O_\mathbbK → O_\mathbbK to be inner. This is done for two different forms of n: (i) n = 2rp (r ∈ ℕ and p an odd rational prime), and (ii) n=pk (k ∈ ℕ ∖ \1\ and p any rational prime). As an application of our main result on classification of (σ, τ)-derivations D:B → 𝔼 and also the conjectures on inner (σ, τ)-derivations of O_\mathbbK, we also conjecture the existence and non-existence of non-zero outer derivations of O_\mathbbK for the above two forms of n, thus answering the twisted derivation problem in O_\mathbbK. Finally, as another application of our main result on the classification of (σ, τ)-derivations D:B → 𝔼, we construct some binary Hom-IDD codes in coding theory.