Alladi Ramakrishnan Hall

Importance of unramified prime ideals of higher residue degree in the study of class groups

Mahesh Kumar Ram

IISER Berhampur

We explore the ideal classes of unramified prime ideals of
residue degree f > 1 in the class groups of number fields. In particular, we have shown that there are many number fields K whose class group is generated by the ideal classes of unramified prime ideals, of K, of residue degree f > 1. As a result of this exploration, we have obtained some nice results on the class groups and class numbers of number fields. For example, if K/Q is a Galois extension of degree 3, then the class number of K cannot be 2. Furthermore, suppose K/Q is an imaginary cyclic quartic extension of number fields. Then, the class number of K cannot be a prime p ≡ 3 (mod 4). By using prime ideals of higher residue degree we have also shown that if K is a number field with the class number greater than 1, then the Hilbert class field of K cannot be a cyclic extension of Q.