Is a finitely generated extension of a real closed field also real closed?

Question

Let $\widetilde{\mathbb{Q}}$ be the field of real algebraic numbers, and consider $\widetilde{\mathbb{Q}}(\pi)$. My question: is $\widetilde{\mathbb{Q}}(\pi)$ a real closed field?

Bonus karma points if you can say something about the general claim: Every finitely generated field extension of a real closed field is also real closed.

Thanks!

Answer 1

Hint: If your field were real-closed, then it would contain $\sqrt[3]{\pi}$. It it did, then $\pi$ would be algebraic.