The isotopy classes of Petit division algebras
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912726228402176 |
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| author | Pumpluen, Susanne |
| author_facet | Pumpluen, Susanne |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18451 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The isotopy classes of Petit division algebras Pumpluen, Susanne Rings and Algebras Information Theory 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. |
| title | The isotopy classes of Petit division algebras |
| topic | Rings and Algebras Information Theory |
| url | https://arxiv.org/abs/2511.18451 |