Lang Undergraduate Algebra

Erscheinungsjahr 2012
ISBN: 978-1-4684-9234-7
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 379 Seiten, eBook

Reihe: Undergraduate Texts in Mathematics

This book, together with Linear Algebra, constitutes a curriculum for an algebra program addressed to undergraduates. The separation of the linear algebra from the other basic algebraic structures fits all existing tendencies affecting undergraduate teaching, and I agree with these tendencies. I have made the present book self contained logically, but it is probably better if students take the linear algebra course before being introduced to the more abstract notions of groups, rings, and fields, and the systematic development of their basic abstract properties. There is of course a little overlap with the book Lin ear Algebra, since I wanted to make the present book self contained. I define vector spaces, matrices, and linear maps and prove their basic properties. The present book could be used for a one-term course, or a year's course, possibly combining it with Linear Algebra. I think it is important to do the field theory and the Galois theory, more important, say, than to do much more group theory than we have done here. There is a chapter on finite fields, which exhibit both features from general field theory, and special features due to characteristic p. Such fields have become important in coding theory.

Zielgruppe

Research

Autoren/Hrsg.

Lang, Serge

Inhaltsverzeichnis

I The Integers.- §1. Terminology of Sets.- §2. Basic Properties.- §3. Greatest Common Divisor.- §4. Unique Factorization.- §5. Equivalence Relations and Congruences.- II Groups.- §1. Groups and Examples.- §2. Mappings.- §3. Homomorphisms.- §4. Cosets and Normal Subgroups.- §5. Permutation Groups.- §6. Cyclic Groups.- §7. Finite Abelian Groups,.- III Rings.- §1. Rings.- §2. Ideals.- §3. Homomorphisms.- §4. Quotient Fields.- IV Polynomials.- §1. Euclidean Algorithm.- §2. Greatest Common Divisor.- §3. Unique Factorization.- §4. Partial Fractions.- §5. Polynomials over the Integers.- §6. Transcendental Elements.- §7. Principal Rings and Factorial Rings.- V Vector Spaces and Modules.- §1. Vector Spaces and Bases.- §2. Dimension of a Vector Space.- §3. Matrices and Linear Maps.- §4. Modules.- §5. Factor Modules.- §6. Free Abelian Group.- VI Some Linear Groups.- §1. The General Linear Group.- §2. Structure of GL2(F).- §3. SL2(F).- VII Field Theory.- §1. Algebraic Extensions.- §2. Embeddings’.- §3. Splitting Fields’.- §4. Galois Theory.- §5. Quadratic and Cubic Extensions.- §6. Solvability by Radicals.- §7. Infinite Extensions.- VIII Finite Fields.- §1. General Structure.- §2. The Frobenius Automorphism.- §3. The Primitive Elements.- §4. Splitting Field and Algebraic Closure.- §5. Irreducibility of the Cyclotomic Equation over Q.- §6. Where Does It All Go? Or Rather, Where Does Some of It Go?.- IX The Real and Complex Numbers.- §1. Ordering of Rings.- §2. Preliminaries.- §3. Construction of the Real Numbers.- §4. Decimal Expansions.- §5. The Complex Numbers.- X Sets.- §1. More Terminology.- §2. Zona’s Lemma.- §3. Cardinal Numbers.- §4. Well-ordering.- §1. The Natural Numbers.- §2. The Integers.- §3. Infinite Sets.