On principal ideal testing in algebraic number fields (Q1092112)

The authors present a technique for determining whether an ideal with known \({\mathbb{Z}}\)-basis in an arbitrary algebraic number field is principal. The algorithm is applied to the problem of determining the cyclotomic numbers of order 7 for a prime \(p\equiv 1\) (mod 7). It is shown that if a septic non-residue (mod p) is known, these numbers can be efficiently computed in O((log p)\({}^ 3)\) binary operations making use of the formulae for the cyclotomic numbers of order 7 of \textit{Ph. A. Leonard} and the reviewer [Proc. Am. Math. Soc. 51, 295-300 (1975; Zbl 0311.12008)].