1. ## Field extension question

I'm trying to answer this question:

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

Does anyone have any ideas?

2. Hi. Can you verify if I'm right?

$\displaystyle \mathbb{Q}\subset\mathbb{Q}(\sqrt{2})\subset\mathb b{Q}(\sqrt[4]{2})$ (normal/normal)
$\displaystyle \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2})\subset\mathbb{Q}(\sqrt[6]{2})$ (not normal/normal)
$\displaystyle \mathbb{Q}\subset\mathbb{Q}(\sqrt{2})\subset\mathb b{Q}(\sqrt[6]{2})$ (normal/not normal)
$\displaystyle \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2})\subset\mathbb{Q}(\sqrt[9]{2})$ (not normal/not normal)

Now assume $\displaystyle M$ is a normal extention over $\displaystyle K,$ and $\displaystyle K\subseteq L\subseteq M$
$\displaystyle M$ is normal over $\displaystyle K$ iff given an algebraic closure $\displaystyle C$ of $\displaystyle M,$ any $\displaystyle K$-isomorphism from $\displaystyle M$ to another subfield of $\displaystyle C$ is a $\displaystyle K$-automorphism of $\displaystyle M.$ Since a $\displaystyle L$-isomorphism from $\displaystyle M$ to another subfield of $\displaystyle C$ is also a $\displaystyle K$-isomorphism, then it is a $\displaystyle L$-automorphism of $\displaystyle M$ and $\displaystyle M$ is a normal extension over $\displaystyle L.$

Finally, consider:

$\displaystyle \mathbb{Q}\subset\mathbb{Q}(j)\subset\mathbb{Q}(\s qrt[3]{2},j)$
$\displaystyle \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2})\subset\mathbb{Q}(\sqrt[3]{2},j)$

$\displaystyle \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2},j)$ is a normal extension, but what about $\displaystyle \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2})$ and $\displaystyle \mathbb{Q}\subset\mathbb{Q}(j)$?

3. Originally Posted by KSM08
I'm trying to answer this question:

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

Does anyone have any ideas?

$\displaystyle K=\mathbb{Q}\leq L=\mathbb{Q}(\sqrt{2})\leq M=\mathbb{Q}(2^{1\slash 4})$ . $\displaystyle M\slash K$ isn't normal, but $\displaystyle M\slash L\,,\,L\slash K$ are.

$\displaystyle K=\mathbb{Q}\leq L=\mathbb{Q}(2^{1\slash 4})\leq M=\mathbb{Q}(2^{1\slash 8})$ . $\displaystyle M\slash K\,,\,L/K$ aren't normal, but $\displaystyle M\slash L$ is.

Now you try other cases.

Tonio