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If $F=K(u_1,\ldots,u_n)$ is a finitely generated (need not be algebraic) extension of $K$ and $M$ is an intermediate field, then $M$ is a finitely generated extension of $K$.

This question was asked here before but I could not understand the solutions. Can anyone write up a detailed solution for me?

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    It would be better if you point to the post which you have found and not understood, and point out *what* you didn't understand in the answers.2012-04-28

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I donot know which parts we are not clear.

We will use the facts about transcendence basis.

Choose a transcendence basis $x_1,\ldots,x_r$ (must be finite) for $M$, then it suffices to show $[M:K(x_1,\ldots,x_r)]$ is finite. Extending $x_1,\ldots,x_r$ to $x_1,\ldots,x_s$ be a transcendce basis of $F$ over $K$. Show $[M:K(x_1,\ldots,x_r)]=[M(x_{r+1},\ldots,x_s):K(x_1,\ldots,x_s)]\leq [F:K(x_1,\ldots,x_s)]<\infty$.