Algebra

This is certainly a classic - but is it any good? This (first) edition is full of typos, has a badly organised index, limited cross-references and very few examples (allegedly it was written entirely linearly by dictation with little editing, and this is wholly believable based only on the text). But it does have a little bit of everything.

Indeholder "Foreword", "Logical Prerequisites", "Bibliography", "Part One. Groups, Rings, and Modules", "Chapter I, Groups", " 1. Monoids", " 2. Groups", " 3. Cyclic groups", " 4. Normal subgroups", " 5. Operations of a group on a set", " 6. Sylow subgroups", " 7. Categories and functors", " 8. Free groups", " 9. Direct sums and free abelian groups", " 10. Finitely generated abelian groups", " 11. The dual group", "Chapter II, Rings", " 1. Rings and homomorphisms", " 2. Commutative rings", " 3. Localization", " 4. Principal rings", "Chapter III, Modules", " 1. Basic definitions", " 2. The group of homomorphisms", " 3. Direct products and sums of modules", " 4. Free modules", " 5. Vector spaces", " 6. The dual space", "Chapter IV, show more Homology", " 1. Complexes", " 2. Homology sequence", " 3. Euler characteristic", " 4. The Jordan-Hölder theorem", "Chapter V, Polynomials", " 1. Free algebras", " 2. Definition of polynomials", " 3. Elementary properties of polynomials", " 4. The Euclidian algorithm", " 5. Partial fractions", " 6. Unique factorization in several variables", " 7. Criteria for irreducibility", " 8. The derivative and multiple roots", " 9. Symmetric polynomials", " 10. The resultant", "Chapter VI, Noetherian Rings and Modules", " 1. Basic criteria", " 2. Hilbert's theorem", " 3. Power series", " 4. Associated primes", " 5. Primary decomposition", "Part Two, Field Theory", "Chapter VII, Algebraic Extensions", " 1. Finite and algebraic extensions", " 2. Algebraic closure", " 3. Splitting fields and normal extensions", " 4. Separable extensions", " 5. Finite fields", " 6. Primitive elements", " 7. Purely inseparable extensions", "Chapter VIII, Galois Theory", " 1. Galois extensions", " 2. Examples and applications", " 3. Roots of unity", " 4. Linear independence of characters", " 5. The norm and trace", " 6. Cyclic extensions", " 7. Solvable and radical extensions", " 8. Kummer theory", " 9. The equation X^n - a = 0", " 10. Galois cohomology", " 11. Algebraic independence of homomorphisms", " 12. The normal basis theorem", "Chapter IX, Extensions of Rings", " 1. Integral ring extensions", " 2. Integral Galois extensions", " 3. Extension of homomorphisms", "Chapter X, Transcendental Extensions", " 1. Transcendence bases", " 2. Hilbert's Nullstellensatz", " 3. Algebraic sets", " 4. Noether normalization theorem", " 5. Linearly disjoint extensions", " 6. Separable extensions", " 7. Derivations", "Chapter XI, Real Fields", " 1. Ordered fields", " 2. Real fields", " 3. Real zeros and homomorphisms", "Chapter XII, Absolute Values", " 1. Definitions, dependence, and independence", " 2. Completions", " 3. Finite extensions", " 4. Valuations", " 5. Completions and valuations", " 6. Discrete valuations", " 7. Zeros of polynomials overcomplete fields", "Part Three, Linear Algebra and Representations", "Chapter XIII, Matrices and Linear Maps", " 1. Matrices", " 2. The rank of a matrix", " 3. Matrices and linear maps", " 4. Determinants", " 5. Duality", " 6. Matrices and bilinear forms", " 7. Sesquilinear duality", "Chapter XIV, Structure of Bilinear Forms", " 1. Preliminaries, orthogonal sums", " 2. Quadratic maps", " 3. Symmetric forms, orthogonal bases", " 4. Hyperbolic spaces", " 5. Witt's theorem", " 6. The Witt group", " 7. Symmetric forms over ordered fields", " 8. The Clifford algebra", " 9. Alternating forms", " 10. The Pfaffian", " 11. Hermitian forms", " 12. The spectral theorem (hermitian case)", " 13. The spectral theorem (symmetric case)", "Chapter XV, Representation of One Endomorphism", " 1. Representations", " 2. Modules over principal rings", " 3. Decomposition over one endomorphism", " 4. The characteristic polynomial", "Chapter XVI, Multilinear Products", " 1. Tensor product", " 2. Basic properties", " 3. Extension of the base", " 4. Tensor product of algebras", " 5. The tensor algebra of a module", " 6. The Alternating Product", " 7. Symmetric products", " 8. The Euler-Grothendieck Ring", " 9. Some functorial isomorphisms", "Chapter XVII, Semisimplicity", " 1. Matrices and linear maps over non-commutative rings", " 2. Conditions defining semisimplicity", " 3. The density theorem", " 4. Semisimple rings", " 5. Simple rings", " 6. Balanced modules", "Chapter XVIII, Representations of Finite Groups", " 1. Semisimplicity of the group algebra", " 2. Characters", " 3. One-dimensional representations", " 4. The space of class functions", " 5. Orthogonality relations", " 6. Induced characters", " 7. Induced representations", " 8. Positive decomposition of the regular character", " 9. Supersolvable groups", " 10. Brauer's theorem", " 11. Field of definition of a representation", "Appendix. The Transcendence of e and π", "Index".

Meget kompakt gennemgang af algebra på universitetsniveau. Stort set ingen eksempler i denne udgave. show less