| Description |
1 online resource (550 pages) |
| Series |
Chapman and Hall/CRC Pure and Applied Mathematics |
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Chapman and Hall/CRC Pure and Applied Mathematics
|
| Contents |
Cover; Half Title; Title Page; Copyright Page; Preface; Table of Contents; 1: Introduction: The art of doing arithmetic; Euler's and Fermat's theorem; Divisibility rules; Other examples for the use of con gruence classes to obtain number-theoretical results; Wilson's theorem; Exercises; 2: Rings and ring homomorphisms; Rings, commutative rings and unital rings; Subrings; Examples; Power seriesrings; Polynomial rings; Matrix rings; Rings of functions; Convolutionrings; Direct products and sums; Ring homomorphisms, isomorphisms and embeddings; Exercises; 3: Integral domains and fields |
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Zero-divisorsNilpotent elements; Units; Examples; Divisibility; Integraldomains; Fields and skew-fields; Quotient fields; Application: Mikusinski'soperator calculus; Exercises; 4: Polynomial and power series rings; Polynomials in one or more variables; Division with remainder; Roots and theirmultiplicities; Derivatives; Symmetric polynomials; Main theorem on symmetricpolynomials; Discriminant; Exercises; 5: Ideals and quotient rings; Ideals; Ideals generated by a set; Principal ideals; Simple rings; Quotientrings; Isomorphism theorems; Maximal ideals; Chinese remainder theorem; Exercises |
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6: Ideals in commutative ringsPrime and primary ideals; Ideal quotients; Radical of an ideal; radical ideals; Primary decompositions; Symbolic powers; Exercises; 7: Factorization in integral domains; Prime and irreducible elements; Expressing notions of divisibility in terms of ideals; Factorization domains and unique factorization domains; Characterization ofprincipal ideal domains; Euclidean domains; Examples; Euclidean algorithm; Exercises; 8: Factorization in polynomial and power series rings; Transmission of the unique factorization property from R to R[x1 ..., xn] |
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Uniquefactorization property for power series ringsFactorization algorithms for polynomials; Irreducibility criteria for polynomials; Resultant of two polynomials; Decomposition of homogeneous polynomials; Exercises; 9: Number-theoretical applications of unique factorization; Representability of prime numbers by quadratic forms; Legendre symbol; Quadratic reciprocity law; Three theorems of Fermat; Ramanujan-Nagell theorem; Insolvability ofthe equation x3+y3 = z3 in N; Kummer's theorem; Exercises; 10: Modules and integral ring extensions; Modules; Free modules; Submodules and quotient modules |
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Module homomorphismsNoetherian and Artinian modules; Algebras over commutativerings; Algebraic and integral ring extensions; Module-theoretical characterizationof integral elements; Integral closure; Exercises; 11: Noetherian rings; Characterization of Noetherian rings; Examples; Hilbert's Basis Theorem; Cohen's theorem; Noetherian induction; Primary decompositions in Noetherianrings; Artinian rings; Exercises; 12: Field extensions; Field extensions; Intermediate fields; Minimal polynomia; Degree of simpleextensions; Degree formula; Liiroth's theorem; Cardinality of finitefields |
| Notes |
Trace and norm of finite field extensions |
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Print version record |
| Subject |
Algebra, Abstract.
|
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Algebra, Abstract
|
| Form |
Electronic book
|
| ISBN |
9781351469258 |
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1351469258 |
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9781351469241 |
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135146924X |
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9781351469234 |
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1351469231 |
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9781315136554 |
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1315136554 |
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