Introduction |
Numbers / Chapter 1: |
Induction / Chapter 2: |
Another Form of Induction / A.: |
Well-Ordering / C.: |
Division Theorem / D.: |
Bases / E.: |
Operations in Base a / F.: |
Euclid's Algorithm / Chapter 3: |
Greatest Common Divisors |
Bezout's Identity |
The Efficiency of Euclid's Algorithm |
Euclid's Algorithm and Incommensurability |
Unique Factorization / Chapter 4: |
The Fundamental Theorem of Arithmetic |
Exponential Notation |
Primes |
Primes in an Interval |
Congruences / Chapter 5: |
Congruence Modulo m |
Basic Properties |
Divisibility Tricks |
More Properties of Congruence |
Linear Congruences and Bezout's Identity |
Congruence Classes / Chapter 6: |
Congruence Classes (mod m): Examples |
Congruence Classes and Z/mZ |
Arithmetic Modulo m |
Complete Sets of Representatives |
Units |
Applications of Congruences / Chapter 7: |
Round Robin Tournaments |
Pseudorandom Numbers |
Factoring Large Numbers by Trial Division |
Sieves |
Factoring by the Pollard Rho Method |
Knapsack Cryptosystems |
Rings and Fields / Chapter 8: |
Axioms |
Z/mZ |
Homomorphisms |
Fermat's and Euler's Theorems / Chapter 9: |
Orders of Elements |
Fermat's Theorem |
Euler's Theorem |
Finding High Powers Modulo m |
Groups of Units and Euler's Theorem |
The Exponent of an Abelian Group |
Applications of Fermat's and Euler's Theorems / Chapter 10: |
Fractions in Base a |
RSA Codes |
2-Pseudoprimes |
Trial a-Pseudoprime Testing |
The Pollard p - 1 Algorithm |
On Groups / Chapter 11: |
Subgroups |
Lagrange's Theorem |
A Probabilistic Primality Test |
Some Nonabelian Groups |
The Chinese Remainder Theorem / Chapter 12: |
The Theorem |
Products of Rings and Euler's [phi]-Function |
Square Roots of 1 Modulo m |
Matrices and Codes / Chapter 13: |
Matrix Multiplication |
Linear Equations |
Determinants and Inverses |
M[subscript n](R) |
Error-Correcting Codes, I |
Hill Codes |
Polynomials / Chapter 14: |
Primitive Roots / Chapter 15: |
Factorization into Irreducible Polynomials |
The Fundamental Theorem of Algebra / Chapter 16: |
Rational Functions |
Partial Fractions |
Irreducible Polynomials over R |
The Complex Numbers |
Root Formulas |
The Fundamental Theorem |
Integrating / G.: |
Derivatives / Chapter 17: |
The Derivative of a Polynomial |
Sturm's Algorithm |
Factoring in Q[x], I / Chapter 18: |
Gauss's Lemma |
Finding Roots |
Testing for Irreducibility |
The Binomial Theorem in Characteristic p / Chapter 19: |
The Binomial Theorem |
Fermat's Theorem Revisited |
Multiple Roots |
Congruences and the Chinese Remainder Theorem / Chapter 20: |
Congruences Modulo a Polynomial |
Applications of the Chinese Remainder Theorem / Chapter 21: |
The Method of Lagrange Interpolation |
Fast Polynomial Multiplication |
Factoring in F[subscript p][x] and in Z[x] / Chapter 22: |
Berlekamp's Algorithm |
Factoring in Z[x] by Factoring mod M |
Bounding the Coefficients of Factors of a Polynomial |
Factoring Modulo High Powers of Primes |
Primitive Roots Modulo m / Chapter 23: |
Polynomials Which Factor Modulo Every Prime |
Cyclic Groups and Primitive Roots / Chapter 24: |
Cyclic Groups |
Primitive Roots Modulo p[superscript e] |
Pseudoprimes / Chapter 25: |
Lots of Carmichael Numbers |
Strong a-Pseudoprimes |
Rabin's Theorem |
Roots of Unity in Z/mZ / Chapter 26: |
For Which a Is m an a-Pseudoprime? |
Square Roots of -1 in Z/pZ |
Roots of -1 in Z/mZ |
False Witnesses |
Proof of Rabin's Theorem |
RSA Codes and Carmichael Numbers |
Quadratic Residues / Chapter 27: |
Reduction to the Odd Prime Case |
The Legendre Symbol |
Proof of Quadratic Reciprocity |
Applications of Quadratic Reciprocity |
Congruence Classes Modulo a Polynomial / Chapter 28: |
The Ring F[x]/m(x) |
Representing Congruence Classes mod m(x) |
Inventing Roots of Polynomials |
Finding Polynomials with Given Roots |
Some Applications of Finite Fields / Chapter 29: |
Latin Squares |
Error Correcting Codes |
Reed-Solomon Codes |
Classifying Finite Fields / Chapter 30: |
More Homomorphisms |
On Berlekamp's Algorithm |
Finite Fields Are Simple |
Factoring x[superscript pn] - x in F[subscript p][x] |
Counting Irreducible Polynomials |
Finite Fields |
Most Polynomials in Z[x] Are Irreducible |
Hints to Selected Exercises |
References |
Index |