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Learn advanced concepts in Algebraic Number Theory in this graduate-level lecture from IMPA's 2025 Summer School. Explore properties of trace and norms before delving into how the ring of integral elements maintains free rank [K:k]. Study algebraic integers, the ring of algebraic integers of a number field, 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 including the analytic class number formula. Drawing from classical texts by Borevich, Shafarevich, Endler, Ribenboim, and Samuel, gain deep insights into fundamental concepts of algebraic number theory through this comprehensive mathematical exploration.
