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The following is a previous question with an additional hypothesis: Let $F_1$, $F_2$ and $K$ be fields of characteristic $0$ such that $F_1\cap K=F_2 \cap K=M$, the extensions $F_i/(F_i \cap K)$ are Galois and algebraic, and $[F_1 \cap F_2 :M]$ is finite. Then is $[F_1 F_2 \cap K:M]$ finite?

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It would be nice to edit in a link to that previous question. –  Gerry Myerson Jun 21 '12 at 0:48
    
@Gerry: I added it. –  Arturo Magidin Jun 21 '12 at 2:58

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Again, no. Let $F_1$ be the splitting field of $\{x^2-3,x^4-3,x^8-3,\ldots\}$ over $\mathbb{Q}$; $F_2$ the splitting field of $\{x^2-5,x^4-5,x^8-5,\ldots\}$; and $K$ the splitting field of $\{x^2-15, x^4-15,x^8-15,\ldots\}$.

Then $M=F_1\cap K = F_2\cap K = F_1\cap F_2$, with the intersection being the splitting field of $\{x^2-1, x^4-1,x^8-1,\ldots\}$ (i.e., adjoining the $2^n$th roots of unity to $\mathbb{Q}$). Since $F_i$ is Galois and algebraic over $\mathbb{Q}$, it is also Galois and algebraic over $F_i\cap K$; $[F_1\cap F_2:M]=1$, but $K\subseteq F_1F_2$, and $[K:M]$ is infinite.

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