The Class Number One Problem for Imaginary Octic Non-CM Extensions of Q

In this paper, we present an alternative method to compute all imaginary octic non-CM fields with class number one, listed in Yamamura’s paper (Acta Arith 86:133–47, 1998). We consider such octic fields as an abelian extension over an appropriate quadratic subfield, whereas Yamaura’s method uses a known structure for quartic fields. In fact, we demonstrate that all such imaginary non-CM octic fields with class number one are subfields of the ray class fields of some imaginary quadratic fields with an appropriate conductor. Using the Pari software, as described in Sect. 3, we determine all imaginary octic non-CM fields with class number one. (The codes are included as an appendix.)