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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?

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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.

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