The additive unit structure of pure quartic complex fields (Q1038629)

The authors describe pure quartic complex fields \(K=\mathbb Q(\root4\of D)\) whose rings of integers are generated by units. They form five infinite families and a finite family of seven elements. The proof provides a way to determine effectively the set of generating units. Moreover, for small integers \(m\) an asymptotic formula is given for the number of non-associated integers \(\alpha\in K\) with \(|N_{K/Q}(\alpha)|\leq x\) which are sums of \(m\) units.