Are all first-order truths of real arithmetic also true of the algebraic reals?

Question

Consider sentences in first-order logic which are true of the structure $(\mathbb R, +, \cdot, <, 0, 1)$, where the symbols have their usual meaning. Is every such sentence also true of $(\mathbb A, +, \cdot, <, 0, 1)$, where $\mathbb A$ is the set of algebraic real numbers?

Answer 1

Yes.

The reason is that the two fields are models of the complete theory of Real-Closed Fields. One of the characterization of real-closed fields is that their algebraic closure is an extension of degree $2$, which is clearly true for both fields.