## Abstract

Given a monic separable polynomial π of degree 2n over an arbitrary field and a scalar α, we define generic algebras H_{π} and A_{απ} for the decomposition of π into a product of two polynomials of degree n and for the factorization απ = ^{2} respectively. We investigate representations of degree 1 or 2 of these generic algebras. Every representation of degree 1 of H_{π} factors through an étale algebra of degree (2nn), whereas A_{απ} has no representation of degree 1. We show that every representation of degree 2 of H_{π} or A_{απ} factors through the Clifford algebra of some quadratic form, pointed or not, and thus obtain a description of the quaternion algebras that are split by the étale algebra F_{π} defined by π of by the function field of the hyperelliptic curve X_{απ} with equation y^{2} = απ(x). We prove that every quaternion algebra split by the function field of X_{απ} is also split by F_{π}, and provide an example to show that a quaternion algebra split by F_{π} may not be split by the function field of any curve X_{απ}.

Original language | English |
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Pages (from-to) | 1807-1826 |

Number of pages | 20 |

Journal | Algebras and Representation Theory |

Volume | 23 |

Issue number | 4 |

DOIs | |

State | Published - 1 Aug 2020 |

### Bibliographical note

Publisher Copyright:© 2019, Springer Nature B.V.

## Keywords

- Clifford algebra
- Hyperelliptic curve
- Pointed quadratic form
- Quaternion algebra
- étale algebra