The question is, if L\K and M\L are normal field extensions then is M\K normal? The answer is false, however, I am not sure how they have got to this, I understand the Tower law to be;
[M:K]=[M:L][L:K]
How would I use this to get false? please may you also give an example, I am studying for my finals and would like as much help as appreciated.
Consider the tower $\mathbb Q\subset \mathbb Q(\sqrt{2})\subset \mathbb Q(\sqrt[4]{2})$.
The first extension is the splitting field of $x^2-2$ and the second is the splitting field of $x^2-\sqrt{2}$ so they are both normal extensions. But the $\mathbb Q$-polynomial $x^4-2$ has some roots in the top extension but not all. For example none of the non-real roots of this polynomial are in the top field.