Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 18I / b

True. If $M/K$ is normal, the finite normal extension as a splitting field criterion writes $M$ as the splitting field over $K$ of some $f\in K[T]$. Since $K\subseteq L\subseteq M$, the same field $M$ is the splitting field of $f$, now regarded as a polynomial over $L$. Hence $M/L$ is normal.

False. Letwhere $\omega$ is a primitive cube root of unity. Then $M$ is the splitting field of $T^3-2$ over $\mathbb{Q}$, so $M/K$ is normal. The irreducible polynomial $T^3-2$ has a root in $L$ but does not split there, so $L/K$ is not normal.

False. TakeEach of $L/K$ and $M/L$ is a quadratic extension and hence normal. However, $T^4-2$ has the root $\sqrt[4]{2}$ in $M$ while its roots $i\sqrt[4]{2}$ and $-i\sqrt[4]{2}$ are absent. Thus $M/K$ is not normal.