# Existence of cyclic extensions ramified at only one prime

For an arbitrary odd prime p is there exists a cyclic extension $$K/\mathbb{Q}$$ of degree p with a prime power discriminant?

• Yes, you can find such extensions as subextensions of cyclotomic extensions $\mathbb{Q}(\zeta_q)$, where $q$ is a prime $\equiv 1 \pmod p$, which exists by Dirichlet's theorem on primes in arithmetic progressions. – Alex B. Oct 11 at 19:32