Let $E/K $ be a finite normal extension, then there exists $p(x)\in K[x] $ s.t. $E$ is the splitting field of $p(x) $.
The proof goes as follows: $E=K(a_1,...,a_n) $ for some $a_1,...,a_n\in \Omega $ where $\Omega $ is the algebraic closure of $K $. Let $\mu_i $ be the minimal polynomial of $a_i $ for $i=1,...,n $ We can send any $a_i $ onto another root of $\mu _i $ through an homomorphism that fixes $K $.
Then the thesis follows.
I don't understand why we can send any $a_i $ onto another root of $\mu _i $.
Note: the definition that we use of normal extension is: $L/K$ is a normal extension if every homomorphism $\phi :K\rightarrow \Omega $ that fixes $K $ sends $L $ onto itself.