Structure of relative genus fields of cubic Kummer extensions

• Published in: Boletín de la Sociedad Matemática Mexicana Vol. 29; no. 3
• Main Authors: Aouissi, Siham, Azizi, Abdelmalek, Ismaili, Moulay Chrif, Mayer, Daniel C., Talbi, Mohamed
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.11.2023; Springer Nature B.V
• Subjects: Conductors; Mathematics; Mathematics and Statistics; Original Article; Relative genus field; Multiplicity of conductors; 3-Rank; 11R29; Pure cubic field; Cubic Kummer extension; Primitive ambiguous principal ideals; 11R20; 11R11; Group of ambiguous ideal classes; 11R37; 11R16; 11R27
• ISSN: 1405-213X; 2296-4495
• DOI: 10.1007/s40590-023-00562-8

Let N = K ( D 3 ) be a cubic Kummer extension of the cyclotomic field K = Q ( ζ 3 ) , containing a primitive cube root of unity ζ 3 , with cube free integer radicand D > 1 . Denote by f the conductor of the abelian extension N / K ,  and by N ∗ the relative genus field of N / K . The aim of the present work is to find out all positive integers D and conductors f such that the genus group Gal N ∗ / N ≅ Z / 3 Z × Z / 3 Z is elementary bicyclic.