Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 18F (Galois Theory)

Define ${\Phi }_n(X)={\prod }_{1\le k\le n,(k,n)=1}(X-e^{2\pi ik/n})$. From $X^n-1={\prod }_{d\mid n}{\Phi }_d(X)$, induction and monic division show ${\Phi }_n\in \mathbb{Z}[X]$. Every ${\Phi }_n$ is irreducible over $\mathbb{Q}$. For prime $p$, ${\Phi }_p(X+1)=((X+1{)}^p-1)/X$ is Eisenstein at $p$, proving the assertion.

For odd $n>1$, the primitive $2n$th roots are exactly the negatives of primitive $n$th roots. Since $\phi (2n)=\phi (n)$, their monic root polynomials satisfy ${\Phi }_{2n}(X)={\Phi }_n(-X)$.

Identify $G\simeq (\mathbb{Z}/15\mathbb{Z}{)}^{\times }\simeq C_4\times C_2$, with $\sigma (\zeta )={\zeta }^2$ and $\tau (\zeta )={\zeta }^{-1}$. Its subgroups areTheir fixed fields, in the same layers, areThe real fields are $\mathbb{Q}$, $\mathbb{Q}(\sqrt{5})$, and $L^{+}$. This is the Galois correspondence.