2013-22 Field automorphisms

Find all field automorphisms of the field of real numbers $$\mathbb{R}$$. (A field automorphism of a field $$F$$ is a bijective map $$\sigma : F \to F$$ that preserves all of $$F$$’s algebraic properties.)

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Let S be the set of non-zero real numbers x such that there is exactly one 0-1 sequence {an} satisfying $$\displaystyle \sum_{n=1}^\infty a_n x^{-n}=1$$. Prove that there is a one-to-one function from the set of all real numbers to S.