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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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.