Let $E$ and $F$ be linearly disjoint fields over a base field $K$ (all contained in an algebraic closure $\overline K$). Suppose there is an extension $M/K$ contained in $EF$.
Is it true that $(M\cap E)(M\cap F)= M$?
I was not able to show it. Does anybode have an idea?
Answer
It is not true, see mihaild's comment.
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