A Concrete Introduction to Higher Algebra
This book is written as an introduction to higher algebra for students with a background of a year of calculus. The first edition of this book emerged from a set of notes written in the 1970sfor a sophomore-junior level course at the University at Albany entitled "Classical Algebra." The o...
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| Format: | eBook |
| Language: | English |
| Published: |
New York, NY
Springer New York
1995, 1995
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| Edition: | 2nd ed. 1995 |
| Series: | Undergraduate Texts in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- E. Error-Correcting Codes, I
- F. Hill Codes
- 14 Polynomials
- 15 Unique Factorization
- A. Division Theorem
- B. Primitive Roots
- C. Greatest Common Divisors
- D. Factorization into Irreducible Polynomials
- 16 The Fundamental Theorem of Algebra
- A. Rational Functions
- B. Partial Fractions
- C Irreducible Polynomials over ?
- D. The Complex Numbers
- E. Root Formulas
- F. The Fundamental Theorem
- G. Integrating
- 17 Derivatives
- A. The Derivative of a Polynomial
- B. Sturm’s Algorithm
- 18 Factoring in ?[x], I
- A. Gauss’s Lemma
- B. Finding Roots
- C. Testing for Irreducibility
- 19 The Binomial Theorem in Characteristic p
- A. The Binomial Theorem
- B. Fermat’s Theorem Revisited
- C. Multiple Roots
- 20 Congruences and the Chinese Remainder Theorem
- A. Congruences Modulo a Polynomial
- B. The Chinese Remainder Theorem
- 21 Applications of the Chinese Remainder Theorem
- A. The Method of Lagrange Interpolation
- D. Sieves
- E. Factoring by the Pollard Rho Method
- F. Knapsack Cryptosystems
- 8 Rings and Fields
- A. Axioms
- B. ?/m?
- C. Homomorphisms
- 9 Fermat’s and Euler’s Theorems
- A. Orders of Elements
- B. Fermat’s Theorem
- C. Euler’s Theorem
- D. Finding High Powers Modulo m
- E. Groups of Units and Euler’s Theorem
- F. The Exponent of an Abelian Group
- 10 Applications of Fermat’s and Euler’s Theorems
- A. Fractions in Base a
- B. RSA Codes
- C. 2-Pseudoprimes
- D. Trial a-Pseudoprime Testing
- E. The Pollard p — 1 Algorithm
- 11 On Groups
- A. Subgroups
- B. Lagrange’s Theorem
- C. A Probabilistic Primality Test
- D. Homomorphisms
- E. Some Nonabelian Groups
- 12 The Chinese Remainder Theorem
- A. The Theorem
- B. Products of Rings and Euler’s ?-Function
- C. Square Roots of 1 Modulo m
- 13 Matricesand Codes
- A. Matrix Multiplication
- B. Linear Equations
- C. Determinants and Inverses
- D. Mn(R)
- 1 Numbers
- 2 Induction
- A. Induction
- B. Another Form of Induction
- C. Well-Ordering
- D. Division Theorem
- E. Bases
- F. Operations in Base a
- 3 Euclid’s Algorithm
- A. Greatest Common Divisors
- B. Euclid’s Algorithm
- C. Bezout’s Identity
- D. The Efficiency of Euclid’s Algorithm
- E. Euclid’s Algorithm and Incommensurability
- 4 Unique Factorization
- A. The Fundamental Theorem of Arithmetic
- B. Exponential Notation
- C. Primes
- D. Primes in an Interval
- 5 Congruences
- A. Congruence Modulo m
- B. Basic Properties
- C. Divisibility Tricks
- D. More Properties of Congruence
- E. Linear Congruences and Bezout’s Identity
- 6 Congruence Classes
- A. Congruence Classes (mod m): Examples
- B. Congruence Classes and ?/m?
- C. Arithmetic Modulo m
- D. Complete Sets of Representatives
- E. Units
- 7 Applications of Congruences
- A. Round Robin Tournaments
- B. Pseudorandom Numbers
- C. Factoring Large Numbers by Trial Division
- B. Fast Polynomial Multiplication
- 22 Factoring in Fp[x] and in ?[x]
- A. Berlekamp’s Algorithm
- B. Factoring in ?[x] by Factoring mod M
- C. Bounding the Coefficients of Factors of a Polynomial
- D. Factoring Modulo High Powers of Primes
- 23 Primitive Roots
- A. Primitive Roots Modulo m
- B. Polynomials Which Factor Modulo Every Prime
- 24 Cyclic Groups and Primitive Roots
- A. Cyclic Groups
- B. Primitive Roots Modulo pe
- 25 Pseudoprimes
- A. Lots of Carmichael Numbers
- B. Strong a-Pseudoprimes
- C. Rabin’s Theorem
- 26 Roots of Unity in ?/m?
- A. For Which a Is m an a-Pseudoprime?
- B. Square Roots of ?1 in ?/p?
- C. Roots of ?1 in ?/m?
- D. False Witnesses
- E. Proof of Rabin’s Theorem
- F. RSA Codes andCarmichael Numbers
- 27 Quadratic Residues
- A. Reduction to the Odd Prime Case
- B. The Legendre Symbol
- C. Proof of Quadratic Reciprocity
- D. Applications of Quadratic Reciprocity
- 28 Congruence Classes Modulo a Polynomial
- A. The Ring F[x]/m(x)
- B. Representing Congruence Classes mod m(x)
- C. Orders of Elements
- D. Inventing Roots of Polynomials
- E. Finding Polynomials with Given Roots
- 29 Some Applications of Finite Fields
- A. Latin Squares
- B. Error Correcting Codes
- C. Reed-Solomon Codes
- 30 Classifying Finite Fields
- A. More Homomorphisms
- B. On Berlekamp’s Algorithm
- C. Finite Fields Are Simple
- D. Factoring xpn — x in Fp[x]
- E. Counting Irreducible Polynomials
- F. Finite Fields
- G. Most Polynomials in Z[x] Are Irreducible
- Hints to Selected Exercises
- References