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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.

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For the intersection to make sense, we must assume that $\bar { \mathbb Q} \subset K$. Then the answer is yes.

Indeed, since $L\subset \mathbb Q(t_1,...,t_n)$ it follows that $L$ is of finite type as a field (this is a non-trivial result!).
But then $L$ is algebraic over $\mathbb Q$ (since $L\subset \bar {\mathbb Q}$) and finitely generated , so that $[K:\mathbb Q]\lt \infty$ .

Edit
As a consequence of Dylan's interesting comment, let me give a reference for the non-trivial result invoked above, since apparently it is absent from most algebra textbooks: Bourbaki, Algebra, Chapter v, §14.7, Corollary 3, page 118.

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    Dear @Marc: as opposed to *finitely generated as a $\mathbb Q$- algebra*. And, yes, your abbreviation is definitely *chic*.2011-12-07