Concrete Abstract Algebra From Numbers to Grobner Bases 1st edition by Niels Lauritzen ISBN 0521534100 978-0521534109

1. Numbers

1.1 The natural numbers and the integers

1.1.1 Well ordering and mathematical induction

1.2 Division with remainder

1.3 Congruences

1.3.1 Repeated squaring – an example

1.4 Greatest common divisor

1.5 The Euclidean algorithm

1.6 The Chinese remainder theorem

1.7 Euler’s theorem

1.8 Prime numbers

1.8.1 There are infinitely many prime numbers

1.8.2 Unique factorization

1.8.3 How to compute φ(n)

1.9 RSA explained

1.9.1 Encryption and decryption exponents

1.9.2 Finding astronomical prime numbers

1.10 Algorithms for prime factorization

1.10.1 The birthday problem

1.10.2 Pollard’s ρ-algorithm

1.10.3 Pollard’s (p − 1)-algorithm

1.10.4 The Fermat–Kraitchik algorithm

1.11 Quadratic residues

1.12 Exercises

2. Groups

2.1 Definition

2.1.1 Groups and congruences

2.1.2 The composition table

2.1.3 Associativity

2.1.4 The first non-abelian group

2.1.5 Uniqueness of neutral and inverse elements

2.1.6 Multiplication by g ∈ G is bijective

2.1.7 More examples of groups

2.2 Subgroups and cosets

2.2.1 Subgroups of Z

2.2.2 Cosets

2.3 Normal subgroups

2.3.1 Quotient groups of the integers

2.3.2 The multiplicative group of prime residue classes

2.4 Group homomorphisms

2.5 The isomorphism theorem

2.6 Order of a group element

2.7 Cyclic groups

2.8 Groups and numbers

2.8.1 Euler’s theorem

2.8.2 Product groups

2.8.3 The Chinese remainder theorem

2.9 Symmetric and alternating groups

2.9.1 Cycles

2.9.2 Simple transpositions and “bubble” sort

2.9.3 The alternating group

2.9.4 Simple groups

2.9.5 The 15-puzzle

2.10 Actions of groups

2.10.1 Conjugacy classes

2.10.2 Conjugacy classes in the symmetric group

2.10.3 Groups of order p[sup(r)]

2.10.4 The Sylow theorems

2.11 Exercises

3. Rings

3.1 Definition

3.1.1 Ideals

3.2 Quotient rings

3.2.1 Quotient rings of Z

3.2.2 Prime ideals

3.2.3 Maximal ideals

3.3 Ring homomorphisms

3.3.1 The unique ring homomorphism from Z

3.3.2 Freshman’s Dream

3.4 Fields of fractions

3.5 Unique factorization

3.5.1 Divisibility and greatest common divisor

3.5.2 Irreducible elements

3.5.3 Prime elements

3.5.4 Euclidean domains

3.5.5 Fermat’s two-square theorem

3.5.6 The Euclidean algorithm strikes again

3.5.7 Prime numbers congruent to 1 modulo 4

3.5.8 Fermat’s last theorem

3.6 Exercises

4. Polynomials

4.1 Polynomial rings

4.1.1 Binomial coefficients modulo a prime number

4.2 Division of polynomials

4.3 Roots of polynomials

4.3.1 Differentiation of polynomials

4.4 Cyclotomic polynomials

4.5 Primitive roots

4.5.1 Decimal expansions and primitive roots

4.5.2 Primitive roots and public key cryptography

4.5.3 Yet another application of cyclotomic polynomials

4.6 Ideals in polynomial rings

4.6.1 Polynomial rings modulo ideals

4.7 Theorema Aureum: the law of quadratic reciprocity

4.8 Finite fields

4.8.1 Existence of finite fields

4.8.2 Uniqueness of finite fields

4.8.3 A beautiful identity

4.9 Berlekamp’s algorithm

4.10 Exercises

5. Gröbner bases

5.1 Polynomials in several variables

5.1.1 Term orderings

5.2 The initial term of a polynomial

5.3 The division algorithm

5.4 Gröbner bases

5.4.1 Hilbert’s basis theorem

5.5 Newton revisited

5.6 Buchberger’s S-criterion

5.6.1 The S-polynomials

5.6.2 The S-criterion

5.7 Buchberger’s algorithm

5.8 The reduced Gröbner basis

5.9 Solving equations using Gröbner bases

5.10 Exercises

Appendix A: Relations

A.1 Basic definitions and properties

A.2 Equivalence relations

A.2.1 Construction of the integers Z

A.2.2 Construction of the rational numbers Q

A.3 Partial orderings

Appendix B: Linear algebra

B.1 Linear independence

B.2 Dimension

References

Index