Advanced Number Theory, by Harvey Cohn ISBN 048664023X
Five out of five stars
This book is essentially a demonstration of the application of abstract algebra to number theory. It opens with the definition of congruence, residue classes, groups and quadratic residues. After some additional work with groups there is material on integral domains followed by the bulk of the book where the work is done on ideals.
Once the basic principles of integral domains and ideals is mentioned, the level of difficulty of the material is quickly ramped up. Significant background in the subject and in number theory are both essential for understanding.
This would be an excellent text for an advanced special topics or graduate level course. The melding of the two fundamental topics of number theory and abstract algebra is a demonstration of how two different disciplines can be combined to advance mathematics.