Author | Kempf, George R. author |
---|---|
Title | Algebraic Structures [electronic resource] / by George R. Kempf |
Imprint | Wiesbaden : Vieweg+Teubner Verlag, 1995 |
Connect to | http://dx.doi.org/10.1007/978-3-322-80278-1 |
Descript | IX, 166 p. online resource |
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 Gaussโ 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 Sylowโs 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 Steinitzโs 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 Complex Case -- 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 Laneโs 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 Zornโs Lemma -- 19.2 Applications -- 19.3 The Axiom of Choice -- 19.4 The Proof of Zornโs Lemma -- 19.5 Well-Ordering -- 19.6 Existence of Algebraically Closed Fields -- 20 Torโs -- 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 Hilbertโs Theorem