The isotopy classes of Petit division algebras

TL;DR

This paper characterizes when Petit division algebras derived from skew polynomial rings are isotopic, linking algebraic properties of polynomials to isotopy classes and providing explicit bounds for finite fields.

Contribution

It establishes a precise criterion for isotopy of Petit division algebras based on polynomial similarity and orbit conditions, refining previous results.

Findings

Two irreducible similar skew polynomials have the same bound.

Petit division algebras are isotopic if their defining polynomials are similar.

Explicit upper bounds for non-isotopic algebras over finite fields are provided.

Abstract

Let $R=K[t;\sigma]$ be a skew polynomial ring, where $K$ is a cyclic Galois field extension of degree $n$ with Galois group generated by $\sigma$. We show that two irreducible similar skew polynomials $f,g\in R$ are similar if and only if they have the same bound. We prove that for two irreducible similar skew polynomials $f,g\in R$ the nonassociative Petit division algebras $R/Rf$ and $R/Rg$ are isotopic. We then refine this result and demonstrate that $f$ and $g$ also yield two isotopic nonassociative Petit algebras $R/Rf$ and $R/Rg$, when the two irreducible polynomials in $F[x]$ that define the minimal central left multiples of $f$ and $g$ have identical degree and lie in the same orbit of some group $G$. For finite field we explicitly compute the upper bound for the number of non-isotopic algebras $R/Rf$ obtained by Lavrauw and Sheekey.