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.