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

Haile, Darrell [Indiana University] Rowen, Louis [Bar-Ilan University] Tignol, Jean-Pierre [UCL]

Given a monic separable polynomial P of degree 2n over an arbitrary field and a scalar a, we define generic algebras H_P and HR_{aP} for the decomposition of P into a product of two polynomials of degree n and for the factorization aP=Q^2 respectively. We investigate representations of degree 1 or 2 of these generic algebras. Every representation of degree 1 of H_P factors through an étale algebra of degree C^n_{2n}, whereas HR_{aP} has no representation of degree 1. We show that every representation of degree 2 of H_P or HR_{aP} 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_P defined by P of by the function field of the hyperelliptic curve X_{aP} with equation y^2=aP(x). We prove that every quaternion algebra split by the function field of X_{aP} is also split by F_P, and provide an example to show that a quaternion algebra split by F_P may not be split by the function field of any curve X_{aP}.