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Learn about ramification in algebraic number theory in this graduate-level mathematics lecture from the 2025 Summer School at IMPA. Explore algebraic integers, the ring of algebraic integers of a number field, bases and discriminants, ideals, prime ideals, class groups, and the finiteness of class groups. Delve into unique factorization, prime ideals, ideal norms, discriminants, different ideals, and ramification theory. Study the fundamental equality, quadratic fields, quadratic reciprocity law, cyclotomic fields, and Dirichlet's unit theorem. Examine zeta functions, L-series of number fields, and the analytic class number formula. Based on foundational works by Borevich & Shafarevich, Endler, Ribenboim, and Samuel, this lecture is part of a comprehensive algebraic number theory course taught in Portuguese by Professor Hossein Movasati.
