What are the

algebraic automorphisms of the

field of

real numbers? Even if we don't demand

continuity, there is only the

identity function. This is in

stark contrast to the field of

complex numbers, where

conjugation is a

continuous automorphism, and (extremely) uncountably many discontinuous automorphisms

exist.

The proof is a fine example of how our ability to define concepts in some language (in this case, the language of the real number field) deeply affects the structure.