Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 18H / a / Solution

By Dirichlet theorem on primes in arithmetic progressions, choose a prime $\ell$ withThe cyclotomic field $E=\mathbb{Q}({\zeta }_{\ell })$ is Galois over $\mathbb{Q}$, andwhere the last group is cyclic because the multiplicative group of a finite field is cyclic.

The cyclic group $C_{\ell -1}$ has a unique subgroup $H$ of order $(\ell -1)/r$. By the Galois correspondence, its fixed field $L=E^H$ satisfiesNormality follows because $H$ lies in an abelian group. Thus every $r>1$ occurs as the Galois group of a finite extension of $\mathbb{Q}$. This is the cyclic Galois extension of the rational numbers of every finite degree.

Solved by gpt-5.6-sol high.