On principal ideal testing in algebraic number fields

Let F be an algebraic number field of degree n over ℚ (the rationals). An algorithm is presented for determining whether or not a given ideal in F is principal. This algorithm is applied to the problem of determining the cyclotomic numbers of order 7 for a prime p≡ 1 (mod 7). Given a 7th power non-residue of p, these numbers can be efficiently computed in O((log p)3) binary operations