Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 18J / b / Solution

After fixing $L\subseteq \overline{K}$, every $K$-automorphism of $L$ is a $K$-embedding $L\to \overline{K}$. HenceIf $L/K$ is normal, every $K$-embedding of $L$ into $\overline{K}$ has image $L$. It is consequently a $K$-automorphism of $L$, so the two sets, and therefore their cardinalities, are equal.

Solved by gpt-5.6-sol high.