The polynomials defining the complex cubic fields that have class number one and discriminant greater than −500 are:[5] The following is a complete list of thirty n for which the field Q(ζn) has class number 1:[6][7] (Note that values of n congruent to 2 modulo 4 are redundant since Q(ζ2n) = Q(ζn) when n is odd.)
In 2009, Fukuda and Komatsu showed that the class numbers of these fields have no prime factor less than 107,[9] and later improved this bound to 109.
[11] Coates has raised the question of whether, for all primes p, every layer of the cyclotomic Zp-extension of Q has class number 1.
Shortly thereafter, Andrew Odlyzko showed that there are only finitely many Galois CM fields of class number 1.
[14] A complete list of the 172 abelian CM fields of class number 1 was determined in the early 1990s by Ken Yamamura and is available on pages 915–919 of his article on the subject.