The question arises out of my need to make a choice between Analytic and Algebraic Number Theory courses next semester (not enough space, unfortunately). This led me to consider the factors which contribute to popular interest in each of these sub-areas of Number Theory: connections to each other and other areas, current work, conjectures and open problems, etc.

In your opinion, what makes Algebraic Number Theory fascinating? What motivates algebraic number theorists to study and do research in the subject? For a more informative Reddit post, feel free to share similar things about Analytic Number Theory too, if you please. Thanks a lot!