Algebraic Number Theory

Serge Lang

The well-known textbook Algebraic Number Theory, written by Lang, has been updated and is currently in its second edition. This course covers all the fundamental concepts of classical algebraic number theory, providing the student with the foundational knowledge required for studying advanced subjects within algebraic number theory, such as cyclotomic fields or modular forms.

The first part of the theory presents several fundamental concepts, such as number fields, ideal classes, ideles and adeles, and zeta functions. In addition, it includes Lang’s proof of a Riemann-Roch theorem in number fields, which was included in the very first edition of the book and was written back in the sixties. This interpretation can now be understood as a forerunner to the Arakelov theory. The second part, “Part II,” is devoted to class field theory. In contrast, the third part, “Part III,” is focused on analytic methods and includes an exposition of Tate’s thesis, the Brauer-Siegel.