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Explore algebraic number theory in this graduate-level mathematics lecture from the 2025 Summer School at IMPA. Delve into algebraic integers, rings of algebraic integers in number fields, bases, and discriminants. Learn about ideals, prime ideals, class groups, and the finiteness of class groups. Master unique factorization, prime ideals, and ideal norms. Study discriminants, differents, and ramification, along with the fundamental equality. Examine quadratic fields, quadratic reciprocity law, and cyclotomic fields. Cover Dirichlet's unit theorem, zeta functions, and L-series of number fields, including the analytic class number formula. Based on foundational texts by Borevich & Shafarevich, Endler, Ribenboim, and Samuel, gain advanced insights into algebraic number theory through this comprehensive 1-hour and 43-minute mathematical exploration.
