ABC Conjecture and the Structure of Algebraic Number Fields

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We have investigated the algebraic structure of algebraic number fields, namely, class groups and unit groups, and related diophantine equations. In A1), we have constructed certain real quadratic fields and quartic fields with explicit fundamental units. These constructions are very useful in number theory, and actually, in B2), we have used these constructions to show the existence of infinite family of algebraic cyclic number fields with prescribed class groups with Y.Kishi. In A2) and B1), we have investigated the positive integer solutions of simultaneous Pell equations, and improved the number of positive integer solutions under the ABC conjecture. We also verified a relation between the fundamental units of certain real quadratic fields and the positive integer solutions of simultaneous Pell equations under the ABC conjecture.We have published the following 3 papers A1),A2),A3) and gave 2 invited lectures B1), B2) at international conferences.A1)On a family of real bicyclic biquadratic fields, CRM Proceedings 36 (2004)A2)On simultaneous Diophantine equations, Acta Arithmetica 108 (2003)A3)On zeta functions associated to finite groups, Advanced Studies in Contemporary Mathematics 4 (2002)B1)On a family of simultaneous Pell equations, The 9^<th> Japan-Korea Joint Seminar on Number theory (2004 Oct. at Kujyu)B2)An infinite family of imaginary cyclic fields of degree p-1 which have ideal class groups of p-ranks greater than one, Yokoi-Chowla Conjecture and Related Problems (2003 Oct. at Nagoya)We note that we have held the above symposium "Yokoi-Chowla Conjecture and Related Problems" at Nagoya University with the help of Prof. Toru Nakahara of Saga University and Prof. Hideo Yokoi of Aichi Gakuin University. The symposium benefited from the scientific grants (140030, 140033). We have published the proceedings of the symposium (ISBN 4-921090-99-8) from Furukawa Total Printing.