# Intersections of finitely generated field extensions are finite?

I was reading the following post at MathOverflow: http://mathoverflow.net/questions/21086/when-are-intersections-of-finitely-generated-field-extensions-finitely-generated/21093

I can't comment there, and I feel this is a lower level question, so I want to ask for clarification here.

In the second proof near the end in the answer given, the author says

Since algebraic independence is of finite character, we may assume that $B$ is finite. Since $L'(B)/K(B)$ is algebraic, we have...

It must be obvious, but why is $L'(B)/K(B)$ algebraic? (I'll try to copy the post over here to make this more self-contained in the mean time.)

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Because $L'/K$ is algebraic. – Brandon Carter Feb 28 '12 at 23:56
Missed that part, I feel dumb. Thanks... – Clary Feb 28 '12 at 23:58