The isotopy classes of Petit division algebras

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.