Example of composition of two normal field extensions which is not normal.

Question

If $F\supset E$ and $E\supset k$ are normal extensions. I want a counter-example where $F\supset k$ is not normal

Answer 1

For seeing Normal extension of a Normal extension is Not Normal...

I feel it would be good choice to see that For a group $G$ and $H\leq K\leq G$ :

$H\unlhd K$ and $K\unlhd G$ but $H\ntrianglelefteq G$

Once you know such a chain of subgroups then, It would be obvious to get an example of Corresponding fields(Fixed fields)

One Basic exmple of such extension is :

$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})\subset \mathbb{Q}(\sqrt[4]{2})$

Answer 2

A sure way to create a field extension is by adjoining roots. What kind of root can we adjoin to ensure normality always? The most obvious answer: square roots. What's a common field which isn't closed under square roots? Let's just pick $\Bbb Q$. Can you form a tower $A/B/\Bbb Q$ where both extensions are obtained by adjoining a square root, but $A/\Bbb Q$ is not normal? Biquadratic $A$ won't cut it (since then it'd be normal). Taking the square root of something in $B=\Bbb Q(\sqrt{b})$ will generally look like $\sqrt{a+\sqrt{b}}$; is there anyway we can make this simpler-looking? What's the simplest type of quadratic surd? Easy: just let $a=0$. Now what do we have?