Some subfields of R have nontrivial field automorphisms. Field automorphisms are important in particular Galois extension s to the theory of field extensions. The category of Riemann surfaces is a bijective biholomorphic map. An automorphism of a differentiable manifold M is a diffeomorphism. Topology are called an automorphism and continuous maps. The inner automorphisms form a normal subgroup of Aut are called outer automorphisms. The same definition holds in algebra and any unital ring.