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Let $E/F$ and $K/F$ be field extensions contained in some common field $L$. Suppose $E/F$ is Galois and that $E \cap K = F$. Let $\alpha \in K$ be algebraic over $F$ with minimal polynomial $f$. Show that $f$ is irreducible over $E$.

So far: I know that $f$ has no roots in $E$, since otherwise it would have all roots in $E$ by normality, whence $\alpha \in E$ (contradicting $E \cap K = F$). But I don't see why $f$ mustn't split in $E$ into irreducible factors.

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  • $\begingroup$ Do you perchance mean "common field $F$"? $\endgroup$ – Robert Lewis May 8 '19 at 0:30
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    $\begingroup$ @RobertLewis I mean that $E, K \subset L$. $\endgroup$ – Freddie May 8 '19 at 0:31
  • $\begingroup$ OK, I get it now. Cheers! $\endgroup$ – Robert Lewis May 8 '19 at 0:31

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