Algebraic Structures

by George R. Kempf

Overview

The laws of composition include addition and multiplication of numbers or functions. These are the basic operations of algebra. One can generalize these operations to groups where there is just one law. The theory of this book was started in 1800 by Gauss, when he solved the 2000 year-old Greek problem about constructing regular n-gons by ruler and compass. The theory was further developed by Abel and Galois. After years of development the theory was put in the present form by E. Noether and E. Artin in 1930. At that time it was called modern algebra and concentrated on the abstract exposition of the theory. Nowadays there are too many examples to go into their details. I think the student should study the proofs of the theorems and not spend time looking for solutions to tricky exercises. The exercises are designed to clarify the theory. In algebra there are four basic structures; groups, rings, fields and modules. We present the theory of these basic structures. Hopefully this will give a good introduction to modern algebra. I have assumed as background that the reader has learned linear algebra over the real numbers but this is not necessary.

Product Details

ISBN-13:	9783528065836
Publisher:	American Mathematical Society
Publication date:	03/23/1998
Series:	VW
Edition description:	Softcover Reprint of the Original 1st 1995 ed.
Pages:	165
Product dimensions:	8.90(w) x 6.40(h) x 0.50(d)

About the Author

Prof. Georg Kempf lehrt an der Johns Hopkins University; Baltimore, Maryland, USA.

Table of Contents

1 Fundamentals of Groups.- 1.1 Sets and Mappings.- 1.2 Groups.- 1.3 Formal Properties of Groups and Homomorphisms.- 1.4 Group Actions.- 1.5 Subgroups and Cosets.- 1.6 Normal Subgroups and Quotient Groups.- 1.7 Exponents and Orders.- 1.8 Permutations.- 1.9 More on Group Actions.- 1.10 Products.- 1.11 A Simpler Definition of a Group.- 2 Fundamentals of rings and fields.- 2.1 Rings.- 2.2 Ideals and Quotient Rings.- 2.3 Integral Domains and Fields.- 2.4 The Integers as a Ring.- 2.5 Principal Ideal and Euclidean Domains.- 2.6 Polynomials.- 2.7 Examples of Fields.- 2.8 Gauss’ Theorem.- 2.9 More Polynomials.- 3 Modules.- 3.1 The Definitions.- 3.2 Bases and Free Modules.- 3.3 Vector Spaces.- 3.4 Modules over a Euclidean Domain.- 3.5 Hom.- 4 A little more group theory.- 4.1 Sylow’s Theorems.- 4.2 1D45D-Groups.- 4.3 Cyclic Finite Groups.- 4.4 Solvable and Simple Groups.- 5 Fields.- 5.1 The Beginning.- 5.2 Degree of Finite Extensions.- 5.3 The Field of Algebraic Elements.- 5.4 Splitting Fields.- 5.5 Existence of Automorphisms.- 5.6 Galois Extensions.- 5.7 Galois Theory.- 5.8 Separable Extensions.- 5.9 Steinitz’s Theorem.- 6 More field theory.- 6.1 The Frobenius.- 6.2 Finite Fields.- 6.3 Roots of Unity.- 6.4 Constructible Numbers.- 6.5 Constructing Regular 1D45B-Gons.- 6.6 Solvable Extensions.- 6.7 Transcendence Degree.- 6.8 The General Equation.- 6.9 Algebraically Closed Fields.- 6.10 Endomorphisms of Vector Spaces.- 7 Modern linear algebra.- 7.1 Tensor Products.- 7.2 Multiple Tensor Products.- 7.3 Graded Rings.- 7.4 The Tensor Algebra.- 7.5 The Symmetric Algebra.- 7.6 The Exterior Algebra.- 7.7 Determinants and Inverses.- 7.8 Characteristic Polynomia.- 7.9 Differential Forms.- 8 Quadratic and alternating forms.- 8.1 Quadratic Forms.- 8.2 The Real Case.- 8.3 The ComplexCase.- 8.4 Hermitian Forms.- 8.5 Alternating Pairings.- 9 Ring and field extensions.- 9.1 Differentials.- 9.2 Decomposition of Tensor Product of Fields.- 9.3 The Normal Basis Theorem.- 9.4 Trace.- 9.5 Theorem 90.- 9.6 Inseparable Extensions.- 9.7 Artin-Schreier Equation.- 10 Noetherian rings and localization.- 10.1 Noetherian Rings.- 10.2 Spec A.- 10.3 Localization.- 10.4 Exact Sequences.- 10.5 Local Rings.- 10.6 Principal Ideal Domains.- 10.7 Nilpotents.- 10.8 Mac Lane’s Criterion.- 11 Dedekind domains.- 11.1 The Definition.- 11.2 Discrete Valuation Rings.- 11.3 The Class Group.- 11.4 Number Theory.- 11.5 Integral Closure.- 11.6 Gaussian Integers.- 12 Representations of Groups.- 12.1 Introduction.- 12.2 Uniqueness of Irreducible Decomposition.- 12.3 Irreducible Representation of a Finite Group.- 12.4 Representation of Abelian Groups.- 12.5 Complex Representations.- 13 More modules.- 13.1 Artin-Rees Lemma.- 13.2 Associated Primes.- 13.3 Primitive Modules.- 13.4 Primary Ideals.- 13.5 Uniqueness.- 13.6 Graded Modules.- 13.7 Prime Divisor.- 14 Categories.- 14.1 The Definition.- 14.2 Examples of Categories.- 14.3 Examples of Functors.- 14.4 Natural Transformations.- 15 Completion.- 15.1 Inverse Limits.- 15.2 Completion of Rings.- 15.3 Completion of Modules.- 15.4 Exactness of Inverse Limits.- 15.5 Noetherianness.- 16 Lie algebra.- 16.1 Introduction.- 16.2 The Universal Enveloping Algebra.- 16.3 Revision.- 17 The Clifford algebra.- 17.1 The Statement.- 17.2 The Proof.- 18 Commutative rings.- 18.1 Dimension.- 18.2 Cohen-Seidenberg Theory.- 18.3 Dimension.- 18.4 Noether Normalization.- 19 Logic.- 19.1 Zorn’s Lemma.- 19.2 Applications.- 19.3 The Axiom of Choice.- 19.4 The Proof of Zorn’s Lemma.- 19.5 Well-Ordering.- 19.6 Existence of Algebraically Closed Fields.- 20Tor’s.- 20.1 Complexes.- 20.2 Definition of Tor.- 20.3 The Proofs.- 20.4 Koszul Complex.- 20.5 Different Resolutions.- 20.6 Efficient Ways to Compute Tor.- 20.7 Hilbert’s Theorem.