Let $\overline{\mathbb{Q}}$ the algebraic closure of $\mathbb{Q}$, and $K$ a field extension of $\mathbb{Q}$ (not necessarily algebraic) such that $[K:\mathbb{Q}]= \infty$.
Let $t_1,...,t_n \in K$, and $L=\mathbb{Q}(t_1,...,t_n) \cap \overline{\mathbb{Q}}$.
Is $L$ a field extension of $\mathbb{Q}$ of finite degree ?
Thanks in advance.