On Quaternion Algebras Split by a Given Extension, Clifford Algebras and Hyperelliptic Curves

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 y² = 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.