Eigenvalue fields of hyperbolic orbifolds

In this paper, we prove that if Γ \Gamma is a non-elementary subgroup of O o ( n , 1 , R ) \mathrm {O}_{\mathrm {o}}(n,1,\mathbb {R}) , with n ≥ 2 n\ge 2 , then the eigenvalue field of Γ \Gamma has infinite degree over Q \mathbb {Q} .