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Galois Theory is one of the most beautiful topics in algebra, as both a mathematical theory and as a wonderful history. This is the first book to detail this history. The book covers the basic material of Galois theory and discusses many related topics (Abelian equations, solvable equations of prime degree, and the casus irreducibilis) not mentioned in most standard treatments of Galois theory. It also describes the rich history of Galois theory, including the work of Lagrange, Gauss, Abel, Galois, Jordan, and Kronecker._ Many books on Galois theory are "thin and elegant", aiming for a_sparse but beautiful presentation of the theory._ This book is a "fat" book on Galois theory in an effort to help_readers to understand not only the elegance of the ideas, but also where they came from_and how they relate to other areas of mathematics._

Galois theory --- 512.62 --- Equations, Theory of --- Group theory --- Number theory --- Fields. Polynomials --- Galois theory. --- 512.62 Fields. Polynomials

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Numbers, Prime --- #WBIB:dd.Lic.L.De Busschere --- 511.6 --- 511.6 Algebraic number fields --- Algebraic number fields --- Prime numbers --- Numbers, Natural

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Singularities (Mathematics) --- Commutative algebra.

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Algebraic geometry --- Differential geometry. Global analysis --- Toric varieties --- Embeddings, Torus --- Torus embeddings --- Varieties, Toric --- Algebraic varieties --- Toric varieties. --- Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra.

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This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem, and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D). The book may serve as a first or second course in undergraduate abstract algebra and, with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple™, Mathematica®, and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used. From the reviews of previous editions: “…The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations, and elimination theory. …The book is well-written. …The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.” —Peter Schenzel, zbMATH, 2007 “I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures, and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry.” —The American Mathematical Monthly.

Mathematics. --- Algebraic Geometry. --- Commutative Rings and Algebras. --- Mathematical Logic and Foundations. --- Mathematical Software. --- Geometry, algebraic. --- Algebra. --- Computer software. --- Logic, Symbolic and mathematical. --- Mathématiques --- Algèbre --- Logiciels --- Logique symbolique et mathématique --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Algebraic geometry. --- Commutative algebra. --- Commutative rings. --- Mathematical logic. --- Geometry, Algebraic --- Commutative algebra --- Data processing --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Mathematical analysis --- Algebraic geometry --- Software, Computer --- Computer systems --- Data processing. --- Rings (Algebra) --- Algebra --- Ordered algebraic structures --- Mathematical control systems --- Computer architecture. Operating systems --- Computer. Automation --- Geometry, Algebraic - Data processing --- Commutative algebra - Data processing --- Geometry, Algebraic.