Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 18H / a / Solution

The composite $L_1{L}_2$ is the smallest subfield of $M$ containing both $L_1$ and $L_2$. Since $L_1/K$ is separable and normal, its minimal polynomials remain separable and split after base change to $L_2$. Hence $L_1{L}_2/L_2$ is Galois.

Restriction definesIt is a homomorphism because every element of the source fixes $L_2$ and therefore $K$. Its kernel fixes both generating fields $L_1$ and $L_2$, hence fixes their composite; the map is injective.