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Is there a isomorphism between the additive group of real algebraic numbers and the multiplicative group of positive real algebraic numbers, which is order preserving.

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up vote 20 down vote accepted

Any such isomorphism would have to extend to a continuous morphism from the additive group of the real numbers to the multiplicative group of the positive real numbers because the order-preserving condition means convergent sequences get sent to convergent sequences. Any such morphism is $a^x$ for some positive real $a$, and this is impossible e.g. by the Gelfond-Schneider theorem.

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+1! This is Cool! –  user977 Aug 10 '10 at 19:16
    
+1 Very clever. –  Akhil Mathew Aug 18 '10 at 11:36

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