Take the 2-minute tour ×
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.

Prove that there exists an isomorphism between the additive group of real, algebraic numbers and the multiplicative group of positive real algebraic numbers. Also, is there one which is order-preserving?

share|improve this question
Ideas, background, self work...something? –  DonAntonio Dec 21 '12 at 9:36
possible duplicate of Isomorphism between algebraic Numbers –  Qiaochu Yuan Dec 21 '12 at 9:37
it is not an exact duplicate as the question in the link requires the isomorphism to be order preserving. –  DonAntonio Dec 21 '12 at 9:53
Why so many downvotes for this question? I've seen hundreds of stupid questions upvoted or ignored by the users, and this one, that is at least interesting, collected 7 downvotes. Dear users, let's think before down(up)voting! –  user26857 Dec 21 '12 at 17:34

1 Answer 1

The set $\mathbb R\cap\overline{ \mathbb Q}$ is a vector space over $\mathbb Q$ by means of the usual addition and multiplication. The set $\mathbb R_{>0}\cap\overline{ \mathbb Q}$ is a vector space over $\mathbb Q$ by means of $(a,b)\mapsto ab$ as addition and $(\frac nm,a)\mapsto \sqrt[m]a^n$ as scalar multiplication. The two vector spaces have the same dimension $\aleph_0$, hence are isomorphic.

For the negative answer in the order preseving case see the link suggested by Qiaochu Yuan.

share|improve this answer

Your Answer


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

Not the answer you're looking for? Browse other questions tagged or ask your own question.