  Jacobian conjectures (that nonsingular implies invertible) for rational
everywhere defined maps of real n-space to itself are considered, with no
requirement for a constant Jacobian determinant or a rational inverse. The
associated extension of rational function fields must be of odd degree and must
have no nontrivial automorphisms. The extensions for the Pinchuk
counterexamples to the strong real Jacobian conjecture have no nontrivial
automorphisms, but are of degree six. The birational case is proved, the Galois
case is clarified but the general case of odd degree remains open. However,
certain topological conditions are shown to be sufficient. Reduction theorems
to specialized forms are proved.
