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Let $E/F$ be a finite Galois extension. Let $K$ be a function field of transcendence degree one over $F$. Let $KE$ be the compositum of $K$ and $E$. Why is $KE/K$ also finite and Galois?

Also, why is $[KE:K]\leq [E:F]$?

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    Also, why is $[KE:K]\leq[E:F]$?2012-03-02
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    Why put the question in a comment, instead of in the body?2012-03-02
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    P.S. What you are considering is better described as a "lift" than a compositum.2012-03-02

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