Title: Number Fields with Absolutely Abelian Hilbert Class Fields
Abstract: The theory of ideal class groups arose from early attempts to understand Fermat’s Last Theorem and has since become a central topic in algebraic number theory. One of the major advances in this area came with class field theory, which identifies the ideal class group of a number field with the Galois group of its Hilbert class field. Although this viewpoint does not make class groups easier to compute, it provides a powerful conceptual framework for studying their structure.
A natural question in this setting is to ask when the Hilbert class field of a number field is absolutely abelian, that is, when it is an abelian extension of $\mathbb{Q}$. In this talk, we present some of our recent results related to this question and also highlight other recent works in which this problem plays an important role. This is joint work with Dr. Prem Prakash Pandey and Dr. Mahesh Kumar Ram.