Normality and Field Extensions

Question

Let M : L : K be finite field extensions. When M is not normal over K, give four examples to show that this gives no information about the normality of M over L or of L over K. What are the possibilities if M is normal over K?

Answer 1

Here's a couple examples. Let $\mathbb{F}$ have characteristic $p$ and consider

1. $\mathbb{F}_p(x):\mathbb{F}_p(x^p):\mathbb{F}$,

2. $\mathbb{F}_p(x,\alpha):\mathbb{F}_p(x):\mathbb{F}(x^p)$

where $\alpha$ is algebraic.

Answer 2

The question being 4 months old, I guess it's safe to assume the homework assignment was due a while ago, so we can give answers.

In all cases, take $K={\bf Q}$.

Let $M={\bf Q}(2^{1/16})$, so $M$ is not normal over $K$.

Take $L_1={\bf Q}(2^{1/8})$, $L_2={\bf Q}(2^{1/4})$, $L_3={\bf Q}(2^{1/2})$.

Then: $M/L_1$ is normal, $L_1/K$ is not. $M/L_2$ is not normal, neither is $L_2/K$. $M/L_3$ is not normal, but $L_3/K$ is. $L_2/K$ is not normal, but $L_2/L_3$ and $L_3/K$ both are.

Still with $K={\bf Q}$:

If $M/K$ is normal, then $M/L$ is normal, but $L/K$ could go either way. E.g., if $M={\bf Q}(\root4\of2,i)$, then $L_1={\bf Q}(\root4\of2)$ is not normal over $K$, but $L_2={\bf Q}(i)$ is.