Showing a counterexample regarding normal extension

Question

For field extensions K/E, E/F, if K/F is a normal extension, E/F is a normal extension also? I think this is false..but can't find a counterexample. Could anyone suggest me some example?

Answer 1

Hint: consider $F=\mathbb{Q}$, $K$ the splitting field of $x^{3}-2$, and $E = F(\sqrt[3]{2})$. $K$ is a normal extension of $F$, and $K$ is a normal extension of $E$ since one can show $K$ is a quadratic extension of $E$. But $E$ is not a normal extension of $F$.

Answer 2

Have some intuition:

An extension $K/F$ is Galois if and only if $K$ is both separable and normal. Further, all extensions of $\mathbb{Q}$ are separable.

If your statement is true, this would imply that all algebraic extensions of $\mathbb{Q}$ are Galois. Equivalently, this would imply that every algebraic extension of $\mathbb{Q}$ is a splitting field for some irreducible polynomial with coefficients over the rationals. Needless to say, this is a contradiction.