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Learn advanced concepts in Algebraic Number Theory in this graduate-level lecture from IMPA's 2025 Summer School. Explore integrally closed rings and minimal polynomials in the first segment, followed by an in-depth discussion of traces and norms. Delve into topics including algebraic integers, rings of algebraic integers of number fields, bases and discriminants, ideals, prime ideals, class groups, unique factorization, ideal norms, ramification, quadratic fields, quadratic reciprocity law, cyclotomic fields, Dirichlet's unit theorem, and zeta functions and L-series of number fields. Drawing from classical texts by Borevich, Shafarevich, Endler, Ribenboim, and Samuel, gain a comprehensive understanding of fundamental concepts in algebraic number theory through this nearly two-hour mathematical exploration.
