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

This is 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.

Solved by gpt-5.6-sol high.