Given a tangent vector field on a finite-dimensional real smooth manifold, its degree (also known as characteristic or rotation ) is, in some sense, an algebraic count of its zeros and gives useful information for its associated ordinary differential equation. When, in particular, the ambient manifold is an open subset U of ℝ m , a tangent vector field v on U can be identified with a map v → : U → ℝ m , and its degree, when defined, coincides with the Brouwer degree with respect to zero of the corresponding map v → . As is well known, the Brouwer degree in ℝ m is uniquely determined by three axioms called Normalization , Additivity , and Homotopy Invariance . Here we shall provide a simple proof that in the context of differentiable manifolds the degree of a tangent vector field is uniquely determined by suitably adapted versions of the above three axioms.