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.

# the field of real numbers has no non-trivial automorphisms (thing)

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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.