UID:
almafu_9960819771702883
Format:
1 online resource (xiv, 227 pages) :
,
digital, PDF file(s).
ISBN:
1-009-01922-8
,
1-009-00413-1
Content:
This book introduces algebraic number theory through the problem of generalizing 'unique prime factorization' from ordinary integers to more general domains. Solving polynomial equations in integers leads naturally to these domains, but unique prime factorization may be lost in the process. To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces, and modules. It was left to Emmy Noether to encapsulate the properties of rings that make unique prime factorization possible, in what we now call Dedekind rings. The book develops the theory of these concepts, following their history, motivating each conceptual step by pointing to its origins, and focusing on the goal of unique prime factorization with a minimum of distraction or prerequisites. This makes a self-contained easy-to-read book, short enough for a one-semester course.
Note:
Title from publisher's bibliographic system (viewed on 01 Aug 2022).
,
Cover -- Half-title -- Title page -- Copyright information -- Dedication -- Contents -- Preface -- Acknowledgments -- 1 Euclidean Arithmetic -- 1.1 Divisors and Primes -- 1.2 The Form of the gcd -- 1.3 The Prime Divisor Property -- 1.4 Irrational Numbers -- 1.5 The Equation x[sup(2)] - 2y[sup(2)] = 1 -- 1.6 Rings -- 1.7 Fields -- 1.8 Factors of Polynomials -- 1.9 Discussion -- 2 Diophantine Arithmetic -- 2.1 Rational versus Integer Solutions -- 2.2 Fermat's Last Theorem for Fourth Powers -- 2.3 Sums of Two Squares -- 2.4 Gaussian Integers and Primes -- 2.5 Unique Gaussian Prime Factorization -- 2.6 Factorization of Sums of Two Squares -- 2.7 Gaussian Primes -- 2.8 Primes that Are Sums of Two Squares -- 2.9 The Equation y[sup(3)] = x[sup(2)] + 2 -- 2.10 Discussion -- 3 Quadratic Forms -- 3.1 Primes of the Form x[sup(2)]+ky[sup(2)] -- 3.2 Quadratic Integers and Quadratic Forms -- 3.3 Quadratic Forms and Equivalence -- 3.4 Composition of Forms -- 3.5 Finite Abelian Groups -- 3.6 The Chinese Remainder Theorem -- 3.7 Additive Notation for Abelian Groups -- 3.8 Discussion -- 4 Rings and Fields -- 4.1 Integers and Fractions -- 4.2 Domains and Fields of Fractions -- 4.3 Polynomial Rings -- 4.4 Algebraic Number Fields -- 4.5 Field Extensions -- 4.6 The Integers of an Algebraic Number Field -- 4.7 An Equivalent Definition of Algebraic Integer -- 4.8 Discussion -- 5 Ideals -- 5.1 "Ideal Numbers" -- 5.2 Ideals -- 5.3 Quotients and Homomorphisms -- 5.4 Noetherian Rings -- 5.5 Noether and the Ascending Chain Condition -- 5.6 Countable Sets -- 5.7 Discussion -- 6 Vector Spaces -- 6.1 Vector Space Basis and Dimension -- 6.2 Finite-Dimensional Vector Spaces -- 6.3 Linear Maps -- 6.4 Algebraic Numbers as Matrices -- 6.5 The Theorem of the Primitive Element -- 6.6 Algebraic Number Fields and Embeddings in [mathbb(C)] -- 6.7 Discussion -- 7 Determinant Theory.
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7.1 Axioms for the Determinant -- 7.2 Existence of the Determinant Function -- 7.3 Determinants and Linear Equations -- 7.4 Basis Independence -- 7.5 Trace and Norm of an Algebraic Number -- 7.6 Discriminant -- 7.7 Discussion -- 8 Modules -- 8.1 From Vector Spaces to Modules -- 8.2 Algebraic Number Fields and Their Integers -- 8.3 Integral Bases -- 8.4 Bases and Free Modules -- 8.5 Integers over a Ring -- 8.6 Integral Closure -- 8.7 Discussion -- 9 Ideals and Prime Factorization -- 9.1 To Divide Is to Contain -- 9.2 Prime Ideals -- 9.3 Products of Ideals -- 9.4 Prime Ideals in Algebraic Number Rings -- 9.5 Fractional Ideals -- 9.6 Prime Ideal Factorization -- 9.7 Invertibility and the Dedekind Property -- 9.8 Discussion -- References -- Index.
Additional Edition:
ISBN 1-316-51895-7
Language:
English
Subjects:
Mathematics
URL:
Volltext
(URL des Erstveröffentlichers)
URL:
https://doi.org/10.1017/9781009004138
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