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Explore advanced concepts in algebraic number theory in this graduate-level lecture from the 2025 Summer School at IMPA, focusing on field and ring extensions by single algebraic elements, discriminants, and computing rings of integers. Delve into fundamental topics including algebraic integers, rings of algebraic integers of number fields, bases and discriminants, ideals, prime ideals, class groups, unique factorization, ideal norms, ramification theory, 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 deep insights into the theoretical foundations of algebraic number theory through this comprehensive mathematical exploration delivered in Portuguese.
