# Minimal polynomial of an element of a splitting field also splits over the splitting field.

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?

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

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