Modern algebra and the rise of mathematical structures

Leo Corry

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.

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[目次]

  • Introduction: Structures in Mathematics.- One: Structures in the Images of Mathematics.- 1 Structures in Algebra: Changing Images.- 1.1 Jordan and Holder: 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 Korper.- 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.

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この本の情報

書名 Modern algebra and the rise of mathematical structures
著作者等 Corry, Leo
出版元 Birkhäuser
刊行年月 c2004
版表示 2nd rev. ed
ページ数 451 p.
大きさ 24 cm
ISBN 3764370025
NCID BA65374713
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言語 英語
出版国 スイス
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