About relation between degree of extension and normality

Question

Hi i know that if $F<E$ is an field extension and $[E:F]=2 $ then it is normal extension.

How can i show that if $[E:F]=k$ for any $k>2 $ doesnt imply E is normal extension. Basically i need counter example for any $k$ Thx :)

Answer 1

Take $\mathbb Q(x)/\mathbb Q(x^k)$. This is not a normal extension for any $k \geq 3$. The normal closure is $\mathbb Q(x,\zeta)$ with $\zeta = \exp(\frac{2\pi i}{k})$ a primitive $k$-th root of unity.