Give an example of a tower of field extensions \[ K\subseteq L\subseteq M \] such that \(L/K\) and \(M/L\) are normal, but \(M/K\) is not normal.
Solution
Take \[ K=\mathbb Q, \qquad L=\mathbb Q(\sqrt2), \qquad M=\mathbb Q(\sqrt[4]{2}). \] Put \(\alpha=\sqrt[4]{2}\), so \(\alpha^2=\sqrt2\).
1. The extension \(L/K\) is normal.
The field \(L=\mathbb Q(\sqrt2)\) is the splitting field over \(\mathbb Q\) of \[ x^2-2=(x-\sqrt2)(x+\sqrt2). \] Hence \(L/K\) is normal.
2. The extension \(M/L\) is normal.
Over \(L\), the element \(\alpha\) is a root of \[ x^2-\sqrt2. \] This polynomial splits in \(M\) as \[ (x-\alpha)(x+\alpha). \] Moreover \(\alpha\notin L\), since \([\mathbb Q(\alpha):\mathbb Q]=4\) by Eisenstein’s criterion applied to \(x^4-2\), whereas \([L:\mathbb Q]=2\). Thus \(M=L(\alpha)\) is the splitting field of \(x^2-\sqrt2\) over \(L\), so \(M/L\) is normal.
3. The extension \(M/K\) is not normal.
The polynomial \[ x^4-2 \] is irreducible over \(\mathbb Q\) by Eisenstein’s criterion at \(2\), so it is the minimal polynomial of \(\alpha\) over \(K\). Its roots are \[ \alpha,-\alpha,i\alpha,-i\alpha. \] But \(M=\mathbb Q(\alpha)\subseteq\mathbb R\), so \(i\alpha\notin M\). Hence the minimal polynomial of \(\alpha\) over \(K\) does not split in \(M\), and therefore \(M/K\) is not normal.
4. Thus normality of field extensions is not transitive.
Combine <1>1–<1>3.