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Informationen zum Autor Thomas Mack is a mathematician, who earned his PhD in the field of combinatorial group theory and hyperbolic groups from the California Institute of Technology. He has worked for more than fifteen years as a researcher in the areas of defense, robotics, and finance. Klappentext "The book is a concise, rigorous, and self-contained survey of undergraduate mathematics that covers the main results of ten areas in the standard curriculum at that level"-- Zusammenfassung A comprehensive survey of undergraduate mathematics, compressing four years of study into one robust overview. In The Math You Need , Thomas Mack provides a singular, comprehensive survey of undergraduate mathematics, compressing four years of math curricula into one volume. Without sacrificing rigor, this book provides a go-to resource for the essentials that any academic or professional needs. Each chapter is followed by numerous exercises to provide the reader an opportunity to practice what they learned. The Math You Need is distinguished in its use of the Bourbaki style—the gold standard for concision and an approach that mathematicians will find of particular interest. As ambitious as it is compact, this text embraces mathematical abstraction throughout, avoiding ad hoc computations in favor of general results. Covering nine areas—group theory, commutative algebra, linear algebra, topology, real analysis, complex analysis, number theory, probability, and statistics—this thorough and highly effective overview of the undergraduate curriculum will prove to be invaluable to students and instructors alike. Inhaltsverzeichnis Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Definitions and Examples . . . . . . . . . . . . . . . . . . 1 1.2 Subgroups and Group Homomorphisms . . . . . . . . . . 4 1.3 Group Constructions . . . . . . . . . . . . . . . . . . . . 8 1.4 The Isomorphism Theorems . . . . . . . . . . . . . . . . 13 1.5 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . 16 1.7 Permutation Groups . . . . . . . . . . . . . . . . . . . . . 20 1.8 p-Groups and the Sylow Theorems . . . . . . . . . . . . . 27 1.9 Solvable and Nilpotent Groups . . . . . . . . . . . . . . . 30 1.10 Free Groups and Presentations . . . . . . . . . . . . . . . 35 1.11 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 39 1.12 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 40 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2 Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.5 Module Constructions . . . . . . . . . . . . . . . . . . . . 60 2.6 Noetherian Modules . . . . . . . . . . . . . . . . . . . . . 63 2.7 Prime and Maximal Ideals . . . . . . . . . . . . . . . . . 66 2.8 Localization . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.9 Gauss’s Lemma . . . . . . . . . . . . . . . . . . . . . . . 76 2.10 Principal Ideal Domains . . . . . . . . . . . . . . . . . . 78 2.11 Field Extensions . . . . . . . . . . . . . . . . . . . . . . . 85 2.12 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.13 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 92 2.14 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 93 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3 Linear Algebr...
