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

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.