# Additive group of Number Theory

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?

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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

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.