Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|cite|improve this question

1 Answer 1

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.

share|cite|improve this answer
+1! This is Cool! – user977 Aug 10 '10 at 19:16
+1 Very clever. – Akhil Mathew Aug 18 '10 at 11:36

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.