I am studying Fundamental Theorem of Algebra.
$\mathbb C$ is algebraically closed
It is enough to prove theorem by showing this statement $1$,
Statement $1$. If $E/\mathbb C$ is finite extension, then $\mathbb C=E$.
And to prove Statement $1$, it is enough to show Statement $2$
Statement $2$. Suppose $E/\mathbb R$ is finite extension, and $K$ be normal closure of $E/\mathbb R$. Then, $K=\mathbb C$.
Now my goal is to show Statement $2$.
Since all fields on above are characteristic $0$, all extensions are separable.
Therefore $K/\mathbb R$ is Galois Extension. And the book states $K/\mathbb R$ is obviously finite extension.
However I couldn't catch why $[K:\mathbb R]<\infty$.