A parametrization of nonassociative Petit algebras $K[t;σ]/K[t:σ](t^m-a)$ up to equivalence
Year: 2026
Type: preprint
Abstract: Let $K/F$ be a cyclic Galois field extension of degree $n$ with $\mathrm{Gal}(K/F)=\langleσ\rangle$. The nonassociative Petit algebras $K[t;σ]/K[t;σ](t^m-a)$ have recently emerged as important tools in coding theory and cryptography, where the applications only require distinct algebras up to isomorphism. Algebras that are isomorphic via a monomial isomorphism of degree one are called equivalent. We present a novel and precise parametrization of the $F$-equivalence classes of these algebras in the key case that $F$ contains a primitive $n$th root of unity and $n\mid m$. We evaluate the parametrization explicitly when $F$ is a nonarchimedean local field, which has applications to the number field case recently used to construct a nonassociative variant of Learning with Errors.
Source: arXiv (Cornell University)
Authors: Nevins, Monica, Pumpluen, Susanne
Institutions: University of Nottingham, University of Ottawa
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223356849