Steinitz classes in quartic fields
Stephen J. Pierce · Proceedings of the American Mathematical Society · 1974
Let $K$ be normal quartic over the rationals. Let $l \equiv 3$ (4) be an odd prime. If the class number of $K$ is even, there is a normal extension $L$ of degree $l$ over $K$ such that the relative discriminant is principal, but $L$ has no relative integral base over $K$.