Quartic fields with large class numbers

Abstract

We show that there exist infinitely many quartic number fields $K$ with large class numbers such that $K∕\mathbf{Q}$ is a Galois extension whose Galois group is isomorphic to a given finite group. Cho and Kim proved that there are infinitely many totally real cyclic extensions over $\mathbf{Q}$ of degree $4$ with large class numbers. We consider all the other cases of quartic Galois extensions.