A Concrete Introduction to Higher Algebra

1. Polynomials

   1. The Fundamental Theorem of Algebra

      • Lindsay Childs

      Pages 136-141

   2. Irreducible Polynomials in ℝ[x]

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      Pages 142-143

   3. Partial Fractions

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      Pages 144-156

   4. The Derivative of a Polynomial

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      Pages 157-159

   5. Sturm’s Algorithm

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      Pages 160-165

   6. Factoring in ℚ[x], I

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      Pages 166-172

   7. Congruences Modulo a Polynomial

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      Pages 173-174

   8. Fermat’s Theorem, II

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      Pages 175-179

   9. Factoring in ℚ[x], II: Lagrange Interpolation

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      Pages 180-184

   10. Factoring in ℤ _p [x]

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       Pages 185-192

   11. Factoring in ℚ[x], III: mod M

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       Pages 193-204

2. Fields

   1. Front Matter

      Pages 205-205

      PDF

   2. Primitive Elements

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      Pages 207-211

   3. Repeating Decimals, II

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      Pages 212-217

   4. Testing for Primeness

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      Pages 218-221

   5. Fourth Roots of One in ℤ _p

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      Pages 222-224

   6. Telephone Cable Splicing

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      Pages 225-228

   7. Factoring in ℚ[x], IV: Bad Examples mod p

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      Pages 229-230

   8. Congruence Classes Modulo a Polynomial: Simple Field Extensions

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      Pages 231-236

   9. Polynomials and Roots

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      Pages 237-241

This book is written as an introduction to higher algebra for students with a background of a year of calculus. The book developed out of a set of notes for a sophomore-junior level course at the State University of New York at Albany entitled Classical Algebra. In the 1950s and before, it was customary for the first course in algebra to be a course in the theory of equations, consisting of a study of polynomials over the complex, real, and rational numbers, and, to a lesser extent, linear algebra from the point of view of systems of equations. Abstract algebra, that is, the study of groups, rings, and fields, usually followed such a course. In recent years the theory of equations course has disappeared. Without it, students entering abstract algebra courses tend to lack the experience in the algebraic theory of the basic classical examples of the integers and polynomials necessary for understanding, and more importantly, for ap­ preciating the formalism. To meet this problem, several texts have recently appeared introducing algebra through number theory.

• algebra
• automorphism
• binomial
• field
• matrices
• matrix
• polynomial
• ring

• Department of Mathematics, SUNY at Albany, Albany, USA

  Lindsay Childs

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