Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 18H (Galois Theory)

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.

The primitive root ${\zeta }_n$ is a zero of the cyclotomic polynomial ${\Phi }_n$, of degree $\phi (n)$. Its minimal polynomial over $K$ divides ${\Phi }_n$, sowhere divisibility also follows from the embedding of the Galois group of the cyclotomic field into $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$. Over $\mathbb{Q}$, Gauss's irreducibility theorem for cyclotomic polynomials says that ${\Phi }_n$ is irreducible. Therefore

Because $\mathrm{lcm}(9,15)=45$ and suitable integral powers of ${\zeta }_9$ and ${\zeta }_{15}$ generate a primitive forty-fifth root,The degree formula givesBoth fields contain $\mathbb{Q}({\zeta }_3)$, which already has degree two, so