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The isotopy classes of Petit division algebras

Susanne Pumpluen

TL;DR

We address the isotopy classification of Petit division algebras arising from twisted polynomial rings $R=K[t;\sigma]$ by irreducible skew polynomials. The main approach links isotopy to bounds and to the action of a group $G$ on irreducible polynomials in $F[x]$, yielding a criterion: irreducible $f,g$ are isotopic iff their bounds share the same orbit under $G$. The paper proves that irreducible $f,g$ with bounds in the same $G$-orbit give isotopic Petit algebras $R/Rf$ and $R/Rg$, and that their associated MRD codes are equivalent; moreover, when the bound degree is maximal, isomorphism can occur. The results generalize Lavrauw–Sheekey to arbitrary base fields and provide explicit orbit-counting formulas over finite fields, giving an upper bound on the number of nonisotopic semifields of fixed dimension.

Abstract

Let $R=K[t;σ]$ be a skew polynomial ring, where $K$ is a cyclic Galois field extension of degree $n$ with Galois group generated by $σ$. 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.

The isotopy classes of Petit division algebras

TL;DR

We address the isotopy classification of Petit division algebras arising from twisted polynomial rings by irreducible skew polynomials. The main approach links isotopy to bounds and to the action of a group on irreducible polynomials in , yielding a criterion: irreducible are isotopic iff their bounds share the same orbit under . The paper proves that irreducible with bounds in the same -orbit give isotopic Petit algebras and , and that their associated MRD codes are equivalent; moreover, when the bound degree is maximal, isomorphism can occur. The results generalize Lavrauw–Sheekey to arbitrary base fields and provide explicit orbit-counting formulas over finite fields, giving an upper bound on the number of nonisotopic semifields of fixed dimension.

Abstract

Let be a skew polynomial ring, where is a cyclic Galois field extension of degree with Galois group generated by . We show that two irreducible similar skew polynomials are similar if and only if they have the same bound. We prove that for two irreducible similar skew polynomials the nonassociative Petit division algebras and are isotopic. We then refine this result and demonstrate that and also yield two isotopic nonassociative Petit algebras and , when the two irreducible polynomials in that define the minimal central left multiples of and have identical degree and lie in the same orbit of some group . For finite field we explicitly compute the upper bound for the number of non-isotopic algebras obtained by Lavrauw and Sheekey.

Paper Structure

This paper contains 8 sections, 18 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonassociative algebras
  4. Twisted polynomial rings
  5. Nonassociative algebras obtained from twisted polynomial rings
  6. Isotopic Petit algebras
  7. The associated MRD codes
  8. Finite fields

Key Result

Proposition 2.1

Let $F_0\subset F$ be any subfield such that $\mathrm{Gal}(K/F_0)$ is abelian. Let $\tau\in \mathrm{Gal}(K/F_0)$. (i) The map $G:R\rightarrow R$ defined via $G(t)=\sum_{i=0}^{m-1} k_i t^i$, $k_i \in K$, extends $\tau$ to an $F_0$-algebra isomorphism $G: K[t;\sigma] \rightarrow K[t;\sigma]$ if and on We write $G=G_{\tau,\alpha}$. (ii) Let $G_{\tau,\alpha}:R\rightarrow R$ be an $F_0$-algebra isomorp

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • Theorem 3.6
  • proof
  • ...and 19 more