If $K$ is an extension field of $F$ such that $[K:F]=2$. Then $K$ is normal?
I know that if $[K:F]=2$ then $K=F(u)$ where $u$ is the root of $f \in F[x]$. But how do you prove that dimension $2$ implies simple (i.e $K=F(u)$).
And $[K:F]$ is finite, and hence $K$ is algebraic over $F$. Then if an irreducible polynomial in $F[x]$ has one root in K, then it splits over $K$, thus it is normal?
Thanks for the help in advance!