Research Projects

Class groups of number fields are one of the most fundamental object in Algebraic Number Theory (and in the study of Diophantine equations). The concept evolved in early attempts to solve Fermat's Last Theorem. Let K be a number field and Cl(K) denote its class group. It is well known that Cl(K) is generated by prime ideals of residue degree one. Also, prime ideals of residue degree one are of full density in the set of all prime ideals of K. Because of these, the role of prime ideals other than prime ideals of residue degree one is not well explored. We want to explore the role of prime ideals of residue degree bigger than one and ramified prime ideals in the study of class groups. In particular we want to show that there are many number fields K whose class group is generated by prime ideals of residue degree f for a fixed positive integer f bigger than one. Annihilators of class groups are very helpful in understanding class groups and in the study of Diophantine equations (especially when not much is known about the involved class group). Annihilators of class groups is a natural output of the project.