The maximal real algebraic field

Question

I want to know what is the maximal real subfield of $\overline{\mathbb{Q}}$. Namely, what is

$$\overline{\mathbb{Q}}\cap\mathbb{R}?$$

For a moment i thought that this was the field of totally real algebraic numbers, but then i realized is a strict contention only (e.g., $\sqrt[3]{2}$)

Answer 1

It's the set of real numbers which are roots of a non-zero polynomial with rational coefficients.