We can express predicate $x = \alpha$ in the interpretation $(\mathbb{R}, +, =, >, \cdot, 0, 1)$ iff $\alpha$ is algebraic. I think that we should use Tarski–Seidenberg theorem (it says that in $(\mathbb{R}, +, =, >, \cdot, 0, 1)$ we can eliminate quantifiers). I thought that $\Leftarrow$ is obvious, but now I'm not sure, because, if $\alpha$ is algebraic, we can take minimal polynomial, but this polynomial can have irrational coefficients and I can't express them. Can you help me, please?
With the help of Levi answer the $\Leftarrow$ became clear, but the opposite isn't so obvious. Maybe we should use the fact that if this predicate is expressable, then we can express it as an semialgebraic set or a formula without quantifiers, but I don't know how to find a contradiction if $\alpha$ isn't algebraic.
By definition of the minimal polynomial, its coefficients are in $\mathbb Q$, so yes this is indeed quite obvious. But you need to pay attention because the minimal polynomial may have multiple roots, so you must somehow say "it's the $k$-th smallest root of this and this polynomial". This is why you need $>$ in the signature.