دانلود کتاب نظریه حلقه و میدان

Topics included: rings, subrings, quotient rings and ring homomorphisms; field of quotients of an integral domain, ideal theory, isomorphism theorems; unique factorization domains, principal ideal domains, Euclidean domains and Gaussian integers; polynomial rings over unique factorization domains, Schönemann-Eisenstein Irreducibility Criterion for unique factorization domains, Perron’s Criterion, Cohn’s Criterion and Osada’s Criterion; Noetherian rings, modules over rings, free modules, finitely generated modules over Euclidean domains, Smith normal form of matrices over Euclidean domains; fields, field extensions, algebraic closure, finite fields; splitting extension fields, separable extension fields, perfect fields, finite normal extension fields; the Fundamental Theorem of Galois Theory and solvability by radicals. Overview and approach: In addition to being an important branch of mathematics in its own right, ring and field theory (Galois theory) is now an essential tool in number theory, geometry, topology, Lie groups, algebraic geometry, differential equations. The main object of study in Galois theory are roots of single variable polynomials. Many ancient mathematics civilizations (Babylonian, Egyptian, Greek, Chinese, Indian, Persian) knew how to solve quadratic equations. Today, most middle school students memorize the “quadratic formula” by heart. While various incomplete methods for solving cubic equations were developed in the ancient world, a general “cubic formula” (as well as a “quartic formula”) was not known until the 16th century Italian school. It was proven by Ruffini and Abel in 1824, that the roots of the general quintic polynomial could not be solvable in terms of nested roots. Galois theory provides a satisfactory explanation for this. More generally, Galois theory is all about symmetries of the roots of polynomials. An essential concept is the field extension generated by the roots of a polynomial, called the splitting field of a polynomial. The philosophy of Galois theory has also impacted other branches of higher mathematics (Lie groups, topology, number theory, algebraic geometry, differential equations). This book will provide a rigorous proof-based modern treatment of the main results of ring and field theory (Galois theory). About this book: This book was originally intended as a textbook for a one-term senior undergraduate (or graduate) course in ring and field theory, or Galois theory. The students are required to have some knowledge on calculus, linear systems, determinants and matrices, and to have taken a first course on abstract algebra. This book can also serve as a reference for professional mathematicians. Earlier drafts of this book were used several times when the author taught MA475 (and MA675), the third course in abstract algebra at Wilfrid Laurier University. When the author prepared these lecture notes he mainly took [F] as his reference. The author tries to make this book self-contained. Readers will be only required to have some knowledge of first and second year university math background. The book contains 241 carefully selected exercise questions of varying difficulty which will allow students to practice their own computational and proof-writing skills. Sample solutions to some exercise questions are provided, from which students can learn to approach and write their own solutions and proofs. Besides standard ones, some of the exercises are new and very interesting. Some are rather hard. It is not a surprise if the reader cannot solve some of the exercises, particularly for first learners.