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Index
Cover
Title Page
Copyright Page
Preface
Introduction
Contents
I Arithmetic, rings and polynomials
1 Arithmetic and the symmetric group
1.1 Reminder of arithmetic
1.1.1 Ring ℤ / nℤ
1.1.2 Groups and Euler’s totient function
1.2 Cryptography
1.2.1 RSA
1.2.1.1 Construction of the keys
1.2.1.2 Encryption of the message
1.2.1.3 Decryption of the message
1.2.2 Elgamal encryption system
1.2.2.1 Construction of the keys
1.2.2.2 Encryption of the message
1.2.2.3 Decryption of the message
1.3 The symmetric group
1.3.1 Permutations and system of generators of Mn
1.3.2 Simplicity of Nn for n ≥ 5
1.4 Exercises
2 Rings and polynomials
2.1 Rings and polynomials
2.1.1 Integral domains and field of fractions
2.1.2 Rings, characteristic, unique factorization domains
2.1.3 Univariate polynomials and Euclidean division
2.1.4 Extended Euclidean algorithm
2.2 Content of a polynomial and irreducibility criteria
2.2.1 Content of a polynomial over a unique factorization domain
2.2.2 Irreducibility criteria
2.3 Resultant of two polynomials
2.4 Multivariate polynomials
2.4.1 Definitions, lexicographic order
2.4.2 Homogeneous polynomials
2.5 Symmetric polynomials
2.5.1 Definitions and elementary symmetric polynomials
2.5.2 Algebraic independence of the elementary symmetric polynomials
2.6 Irreducibility of certain “general” polynomials
2.6.1 Irreducibility of the general determinant and permanent
2.6.2 Irreducibility of the general resultant
2.7 Exercises
II Galois theory
3 Algebraic extensions
3.1 Algebraic extensions
3.1.1 Algebraic elements
3.1.2 Algebraic extensions: definition and first properties
3.2 Algebraic closures
3.3 Rupture fields, conjugated elements, splitting field
3.4 Existence and uniqueness of finite fields
3.5 Exercises
4 Normal extensions and separable extensions
4.1 Normal extensions
4.2 Separable extensions
4.2.1 Definitions and properties
4.2.2 Primitive element theorem
4.3 Perfect fields
4.4 Exercises
5 Galois theory
5.1 Galois extensions
5.2 Artin theorem
5.3 Fundamental theorem of Galois theory
5.4 Exercises
6 Abelian, cyclic, cyclotomic, radical extensions
6.1 Abelian, cyclic, cyclotomic, radical extensions
6.1.1 Primitive roots of unity
6.1.2 Definition and properties of various extensions and examples
6.1.3 Link between cyclic extensions and radical extensions
6.2 Cyclotomic polynomials
6.2.1 Irreducibility of Φn over ℚ
6.2.2 Factorization of Φn, Fp over Fq
6.3 Exercises
7 Galois group of a polynomial
7.1 Reminder of group theory
7.1.1 p-groups and Sylow theorem
7.1.2 Solvable groups and nilpotent groups
7.2 Galois group of a polynomial
7.3 Discriminant of a polynomial
7.4 Solvable equations
7.5 Transitive subgroups of Mp and applications
7.6 Determining the Galois group of a polynomial
7.6.1 Van der Waerden theorem
7.6.2 Solving equations of degree 2, 3, 4 over ℂ
7.6.2.1 Equations of degree 2
7.6.2.2 Equations of degree 3 — Cardano’s formulas
7.6.2.3 Equations of degree 4 — Ferrari’s formulas
7.6.3 Determining the Galois group of a polynomial of degree less than or equal to 4
7.6.3.1 Polynomial of degree 2
7.6.3.2 Polynomial of degree 3
7.6.3.3 Polynomial of degree 4
7.6.4 Example: crossed ladders problem
7.7 Realizing a group as a Galois group
7.8 Exercises
III Applications
8 Ruler and compass constructions
8.1 Constructible points and numbers
8.2 The Wantzel theorem
8.3 Historical problems
8.4 The Galois correspondence and the regular polygons
8.5 Exercises
9 Finite fields and applications
9.1
9.2 Irreducible polynomials over finite fields
9.2.1 Description of the irreducible polynomials
9.2.2 Explicit construction of finite fields
9.2.2.1 First construction
9.2.2.2 Second construction
9.2.2.3 Examples
9.2.2.4 Construction of an extension of Fq
9.3 Factorization of polynomials of Fq[X]
9.3.1 Definition of the Berlekamp subalgebra
9.3.2 Irreducibility and Berlekamp subalgebra
9.3.3 Factorization method
9.3.4 Berlekamp’s algorithm
9.3.5 Type of factorization modulo p
9.4 Error-correcting codes
9.4.1 Definition and properties of error-correcting codes
9.4.2 Construction of BCH codes
9.4.2.1 BCH codes
9.4.2.2 Reed-Solomon codes
9.4.3 Error detection and correction for the BCH codes
9.4.3.1 Error detection
9.4.3.2 Error correction
9.4.4 Binary BCH codes
9.4.4.1 Construction of the word to send
9.4.4.2 Example
9.4.5 Coding of the compact disks
9.4.5.1 From sound to CD
9.4.5.2 The CIRC coding
9.4.5.3 Decoding
9.5 Exercises
10 Norm, trace and algebraic integers
10.1 Norm and trace
10.1.1 Properties of the norm and the trace
10.1.2 Hilbert’s theorem 90 and Artin-Schreier theorem
10.2 Algebraic integers
10.2.1 Definition and properties
10.2.2 Examples of algebraic integers
10.2.3 Discriminant of a family
10.2.4 Normal basis theorem
10.3 Exercises
Biography of quoted mathematicians
Index
References
About the Author
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