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

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    $\begingroup$ 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. $\endgroup$ – Alex B. Oct 11 at 19:32

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