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AuthorLang, Serge. author
TitleUndergraduate Algebra [electronic resource] / by Serge Lang
ImprintNew York, NY : Springer New York, 1987
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Descript IX, 379 p. 1 illus. in color. online resource


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 Linยญ 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


I The Integers -- ยง1. Terminology of Sets -- ยง2. Basic Properties -- ยง3. Greatest Common Divisor -- ยง4. Unique Factorization -- ยง5. Equivalence Relations and Congruences -- II Groups -- ยง1. Groups and Examples -- ยง2. Mappings -- ยง3. Homomorphisms -- ยง4. Cosets and Normal Subgroups -- ยง5. Permutation Groups -- ยง6. Cyclic Groups -- ยง7. Finite Abelian Groups, -- III Rings -- ยง1. Rings -- ยง2. Ideals -- ยง3. Homomorphisms -- ยง4. Quotient Fields -- IV Polynomials -- ยง1. Euclidean Algorithm -- ยง2. Greatest Common Divisor -- ยง3. Unique Factorization -- ยง4. Partial Fractions -- ยง5. Polynomials over the Integers -- ยง6. Transcendental Elements -- ยง7. Principal Rings and Factorial Rings -- V Vector Spaces and Modules -- ยง1. Vector Spaces and Bases -- ยง2. Dimension of a Vector Space -- ยง3. Matrices and Linear Maps -- ยง4. Modules -- ยง5. Factor Modules -- ยง6. Free Abelian Group -- VI Some Linear Groups -- ยง1. The General Linear Group -- ยง2. Structure of GL2(F) -- ยง3. SL2(F) -- VII Field Theory -- ยง1. Algebraic Extensions -- ยง2. Embeddingsโ{128}{153} -- ยง3. Splitting Fieldsโ{128}{153} -- ยง4. Galois Theory -- ยง5. Quadratic and Cubic Extensions -- ยง6. Solvability by Radicals -- ยง7. Infinite Extensions -- VIII Finite Fields -- ยง1. General Structure -- ยง2. The Frobenius Automorphism -- ยง3. The Primitive Elements -- ยง4. Splitting Field and Algebraic Closure -- ยง5. Irreducibility of the Cyclotomic Equation over Q -- ยง6. Where Does It All Go? Or Rather, Where Does Some of It Go? -- IX The Real and Complex Numbers -- ยง1. Ordering of Rings -- ยง2. Preliminaries -- ยง3. Construction of the Real Numbers -- ยง4. Decimal Expansions -- ยง5. The Complex Numbers -- X Sets -- ยง1. More Terminology -- ยง2. Zonaโ{128}{153}s Lemma -- ยง3. Cardinal Numbers -- ยง4. Well-ordering -- ยง1. The Natural Numbers -- ยง2. The Integers -- ยง3. Infinite Sets

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