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In a proof of the Fundamental Theorem of Galois Theory, my professor used the unproved fact that, if $K$ is the splitting field of some irreducible polynomial $f$ over the field $F$, and if $\alpha\in K$, then the minimal polynomial of $\alpha$ over $F$ also splits completely over $K$. Why is this true?

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  • $\begingroup$ $\alpha \in F$? $\endgroup$ – WWK May 8 '14 at 3:01
  • $\begingroup$ You mean $\alpha\in K$. The standard argument is to look at the orbit of $\alpha$ under the Galois group $G(K/F)$. $\endgroup$ – Ted Shifrin May 8 '14 at 3:12
  • $\begingroup$ Yeah. Could you elaborate a bit? $\endgroup$ – Nishant May 8 '14 at 3:13
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This proof comes from Serge Lang's Algebra.

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  • $\begingroup$ What is $k^a$? Hi. $\endgroup$ – Nishant May 8 '14 at 4:27
  • $\begingroup$ @Nishant: It is an algebraic closure ok $k$, see the second line of theorem $3.3$ $\endgroup$ – WLOG May 8 '14 at 6:21

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