HUDU

Modern Algebra and the Rise of Mathematical Structures


€ 69,99
 
kartoniert
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November 2003

Beschreibung

Beschreibung

This book describes two stages in the historical development of the notion of mathematical structures: first, it traces its rise in the context of algebra from the mid-1800s to 1930, and then considers attempts to formulate elaborate theories after 1930 aimed at elucidating, from a purely mathematical perspective, the precise meaning of this idea.

Inhaltsverzeichnis

Introduction: Structures in Mathematics.- One: Structures in the Images of Mathematics.- 1 Structures in Algebra: Changing Images.- 1.1 Jordan and Hölder: Two Versions of a Theorem.- 1.2 Heinrich Weber:Lehrbuch der Algebra.- 1.3 Bartel L. van der Waerden:Moderne Algebra.- 1.4 Other Textbooks of Algebra in the 1920s.- 2 Richard Dedekind: Numbers and Ideals.- 2.1 Lectures on Galois Theory.- 2.1 Algebraic Number Theory.- 2.2.1 Ideal Prime Numbers.- 2.2.2 Theory of Ideals: The First Version (1871).- 2.2.3 Later Versions.- 2.2.4 The Last Version.- 2.2.5 Additional Contexts.- 2.3 Ideals andDualgruppen.- 2.4 Dedekind and the Structural Image of Algebra.- 3 David Hilbert: Algebra and Axiomatics.- 3.1 Algebraic Invariants.- 3.2 Algebraic Number Theory.- 3.2 Hilbert's Axiomatic Approach.- 3.4 Hilbert and the Structural Image of Algebra.- 3.5 Postulational Analysis in the USA.- 4 Concrete and Abstract: Numbers, Polynomials, Rings.- 4.1 Kurt Hensel: Theory ofp-adicNumbers.- 4.2 Ernst Steinitz:Algebraische Theorie der Körper.- 4.3 Alfred Loewy:Lehrbuch der Algebra.- 4.4 Abraham Fraenkel: Axioms forp-adicSystems.- 4.5 Abraham Fraenkel: Abstract Theory of Rings.- 4.6 Ideals and Abstract Rings after Fraenkel.- 4.7 Polynomials and their Decompositions.- 5 Emmy Noether: Ideals and Structures.- 5.1 Early Works.- 5,2Idealtheorie in Ringbereichen.- 5.3Abstrakter Aufbau der Idealtheorie.- 5.4 Later Works.- 5.5 Emmy Noether and the Structural Image of Algebra.- Two: Structures in the Body of Mathematics.- 6 Oystein Ore: Algebraic Structures.- 6.1 Decomposition Theorems and Algebraic Structures.- 6.2 Non-Commutative Polynomials and Algebraic Structure.- 6.3 Structures and Lattices.- 6.4 Structures in Action.- 6.5 Universal Algebra, Model Theory, Boolean Algebras.- 6.6 Ore's Structures and the Structural Image of Algebra.- 7 Nicolas Bourbaki: Theory ofStructures.- 7.1 The Myth.- 7.2 Structures and Mathematics.- 7.3Structuresand the Body of Mathematics.- 7.3.1 Set Theory.- 7.3.2 Algebra.- 7.3.3 General Topology.- 7.3.4 Commutative Algebra.- 7.4Structuresand the Structural Image of Mathematics.- 8 Category Theory: Early Stages.- 8.1 Category Theory: Basic Concepts.- 8.2 Category Theory: A Theory of Structures.- 8.3 Category Theory: Early Works.- 8.4 Category Theory: Some Contributions.- 8.5 Category Theory and Bourbaki.- 9 Categories and Images of Mathematics.- 9.1 Categories and the Structural Image of Mathematics.- 9.2 Categories and the Essence of Mathematics.- 9.3 What is Algebra and what has it been in History?.- Author Index.
EAN: 9783764370022
ISBN: 3764370025
Untertitel: 2nd revised ed. 2004. Book. Sprache: Englisch.
Verlag: Birkhäuser
Erscheinungsdatum: November 2003
Seitenanzahl: 472 Seiten
Format: kartoniert
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