On class numbers of pure quartic fields

Abstract

Let p be a prime. The 2-primary part of the class group of the pure quartic field $${\mathbb {Q}}(\root 4 \of {p})$$ has been determined by Parry and Lemmermeyer when $$p \not \equiv \pm \, 1\bmod 16$$ . In this paper, we improve the known results in the case $$p\equiv \pm \, 1\bmod 16$$ . In particular, we determine all primes p such that 4 does not divide the class number of $${\mathbb {Q}}(\root 4 \of {p})$$ .We also conjecture a relation between the class numbers of $${\mathbb {Q}}(\root 4 \of {p})$$ and $${\mathbb {Q}}(\sqrt{-2p})$$ . We show that this conjecture implies a distribution result of the 2-class numbers of $${\mathbb {Q}}(\root 4 \of {p})$$ .