Table of contents
1
2
3
4
5
6
7
8
Chapter 1. Arithmetic
Divisors and multiples
Divisors
Division with remainder
Common divisors
Common multiples
Euclid's algorithm
Euclid's algorithm
Extended Euclidean algorithm
Characterization of gcd
Relatively prime
Linear diophantine equations
Homogeneous equation
Linear equation
Prime numbers
The notion
Eratosthenes' sieve
Characterization of primes
Factorization
The notion
Gcd and lcm
Exercises
Summary of Chapter 1
Chapter 2. Modular arithmetic
Arithmetic modulo
n
Modular congruence
Addition and multiplication
Multiplicative inverse
Linear congruences
The notion
Solving linear congruences
Chinese remainder theorem
The
a
-ary number system
The notion
The RSA cryptosystem
Fermat's little theorem
RSA ingredients
Encoding and decoding
How to use RSA
Radar detection
Detection patterns
Primitive elements
Pattern construction
Exercises
Summary of Chapter 2
Chapter 3. Polynomials
The notion of a polynomial
The notion
Sum and product
Polynomial division
Leading term and degree
Division with remainder
Gcd and lcm
Euclid's algorithm
Characterization of the gcd
Polynomial functions
The notion
Interpolation
Factorization
Irreducible polynomial
Relatively prime
Unique factorization
Shift registers
The notion
Output formula
Linear algebra
Exercises
Summary of Chapter 3
Chapter 4. Arithmetic modulo polynomials
Congruence modulo a polynomial
The notion
Residue class
Class representative
The residue class ring
Construction
Arithmetical rules
Constants modulo a polynomial
Class rings as vector spaces
Projection onto the residue class ring
Two special cases
Approximations
Modulo
n
Inverses and fields
Inverses
Fields
Finite fields
Construction
Properties
Error-correcting codes
Coding theory
Linear codes
Construction
Decoding for cyclic codes
BCH bound for cyclic codes
Exercises
Summary of Chapter 4
Chapter 5. Permutations
Symmetric groups
Maps
Permutation
Notation for permutations
Cycles
Fix and support
Cycles
Cycle structure
Conjugation
Transpositions
Alternating groups
Sign
Sign is multiplicative
Alternating group
3-cycles
Exercises
Summary of Chapter 5
Chapter 6. Monoids and groups
Binary operations
Arity
Associativity
Semi-groups
Monoids
Commutativity
Monoids
Direct product
Submonoid
Free monoid
Morphisms
Cyclic monoids
Invertibility in monoids
Inverse
Properties of inverse
Euler indicator
Groups
Subgroups
Subgroups generated by subsets
Intersections
Some special subgroups
Morphisms
Kernel and image
Cyclic groups
Order of an element
Characterizing cyclic groups
Cosets
The notion
Lagrange
Normal subgroups
Exercises
Summary of Chapter 6
Chapter 7. Rings and fields
The structure ring
Rings and subrings
Some computational laws
Invertible elements
Morphisms
Constructions with rings
Cartesian products
Generation
Polynomial rings
Domains and fields
The notion
More on domains
Fields
Subfields
Field of fractions
Fields
Characteristics
Vector spaces
General properties
Morphisms
Algebraic numbers
Ideals
The notion
Generation
Constructions
Prime and maximal ideals
Residue class rings
Residue class rings
First isomorphism theorem
Prime and maximal ideals
Finite fields
Characterization
Primitive elements
Existence
Hadamard matrices
Exercises
Summary of Chapter 7
Chapter 8. Permutation groups
Permutation groups
The notion
Each group is a permutation group
Constructions
Orbits
The notion
Stabilizers
Transitive actions are actions on cosets
Order
Basis
Schreier trees
Stabilizer determination
Automorphisms
The notion
Finite fields
Quotient groups
The notion
First isomorphism theorem
Small groups
Preliminaries
Order at most 11
Exercises
Summary of Chapter 8