On the rational real Jacobian conjecture

Jacobian conjectures (that nonsingular implies a global inverse) for rational everywhere defined maps of $\bbb{R}^n$ to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The birational case is proved and the Galois case clarified. Two known special cases of the Strong Real Jacobian Conjecture (SRJC) are generalized to the rational map context. For an invertible map, the associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. That disqualifies the Pinchuk counterexamples to the SRJC as candidates for invertibility