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Let $F$ be a field, and $K$ a finite extension of $F$. Suppose that for every irreducible polynomial $P(x)\in F[x]$, if $P(x)$ has one root in $K$, then $P(x)$ has all its roots in $K$. How can we show that $K$ is a splitting field of some polynomial in $F$?

Since $K$ is a finite extension of $F$, there exists $c\in K$ such that $K=F(c)$. Let $P(x)$ be the minimal polynomial of $c$ over $F$. I think $K$ is a splitting field of $P$, but how to show it?

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Nothing left to prove. $P(x)$ is irreducible over $F$, $\,K$ contains all its roots, and $K$ is the extension with (one of) them over $F$.

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  • $\begingroup$ Thanks. You mean $P(x)$ is irreducible over $F$? $\endgroup$ – Kunal Jan 19 '14 at 3:46
  • $\begingroup$ Yes, corrected, thanks. Anyway, the hard part of this question is that 'Since $K$ is a finite extension of $F$, there exists $c\in K$ such that $K=F(c)$.' $\endgroup$ – Berci Jan 20 '14 at 1:28

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