Dense and algebraically independent subset of $\mathbb{R}$ in $(\mathbb{R}, <, +, ·, 0, 1)$.

Question

I was trying to construct a dense subset $D$ of $\mathbb{R}$ that was also algebraically independent, that is, for any $x\in D$ $(x\notin acl(D\cup\{0,1\}\setminus \{x\}))$. I know they exist in saturated extensions of $(\mathbb{R}, <, +, ·, 0, 1)$. However, the usual examples of dense subsets of $\mathbb{R}$ that come to my mind are not algebraically independent. Is there any known example of such a set with an easy description?

Answer 1

The seqeuence $\sqrt{p_k}$ (squareroots of the primes) is linearly independent over $\mathbb Q$. By the Lindemann-Weierstrass theorem, the sequence $\exp(\sqrt{p_k})$ is algebraically independent over $\mathbb Q$. We may choose nonzero rationals $r_k$ so that $r_k\exp(\sqrt{p_k})$ is dense in $\mathbb R$; this sequence is still algebraically independent.