Galois Theory

Galois Theory

Galois Theory is an excellent book for courses on abstract algebra at the upper-undergraduate and graduate levels. The book also serves as an interesting reference for anyone with a general interest in Galois theory and its contributions to the field of mathematics.

1.1 CARDAN’S FORMULAS

1.2 PERMUTATIONS OF THE ROOTS

1.3 CUBIC EQUATIONS OVER THE REAL NUMBERS

2.1 POLYNOMIALS OF SEVERAL VARIABLES

2.2 SYMMETRIC POLYNOMIALS

2.3 COMPUTING WITH SYMMETRIC POLYNOMIALS

2.4 THE DISCRIMINANT

3.1 THE EXISTENCE OF ROOTS

3.2 THE FUNDAMENTAL THEOREM OF ALGEBRA

4.1 FIELDS

4.2 IRREDUCIBLE POLYNOMIALS

4.3 THE DEGREE OF AN EXTENSION

4.4 ALGEBRAIC EXTENSIONS

5.1 NORMAL AND SEPARABLE EXTENSIONS

5.2 NORMAL EXTENSIONS

5.3 SEPARABLE EXTENSION

5.4 THEOREM OF THE PRIMITIVE ELEMENT

6.1 DEFINITION OF THE GALOIS GROUP

6.2 GALOIS GROUPS OF SPLITTING FIELDS

6.3 PERMUTATION OF THE ROOTS

6.4 EXAMPLES OF GALOIS GROUPS

6.5 ABELIAN EQUATION

7.1 GALOIS EXTENSIONS

7.2 NORMAL SUBGROUPS AND NORMAL EXTENSIONS

7.3 THE FUNDAMENTAL THEOREM OF GALOIS THEORY

7.4 FIRST APPLICATIONS

7.5 AUTOMORPHISMS AND GEOMETRY

8.1 SOLVABLE GROUPS

8.2 RADICAL AND SOLVABLE EXTENSIONS

8.3 SOLVABLE EXTENSIONS AND SOLVABLE GROUPS

8.4 SIMPLE GROUPS

8.5 SOLVING POLYNOMIALS BY RADICALS

8.6 THE CASUS IRREDUCIBILIS

9.1 CYCLOTOMIC POLYNOMIALS

9.2 GAUSS AND ROOTS OF UNITY

10.1 CONSTRUCTIBLE NUMBERS

10.2 REGULAR POLYGONS AND ROOTS OF UNITY

10.3 ORIGAMI

11.1 THE STRUCTURE OF FINITE FIELDS

11.2 IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS

12.1 LAGRANGE

12.2 GALOIS

12.3 KRONECKER

13.1 QUARTIC POLYNOMIALS

13.2 QUINTIC POLYNOMIALS

13.3 RESOLVENTS

13.4 OTHER METHODS

14.1 POLYNOMIAL OF PRIME DEGREE

14.2 IMPRIMITIVE POLYNOMIALS OF PRIME-SQUARED DEGREE

14.3 PRIMITIVE PERMUTATION GROUPS

14.4 PRIMITIVE POLYNOMIALS OF PRIME-SQUARED DEGREE

15.1 DIVISION POINTS AND ARC LENGTH

15.2 THE LEMNISCATIC FUNCTION

15.3 THE COMPLEX LEMNISCATIC FUNCTION

15.4 COMPLEX MULTIPLICATION

15.5 ABEL’S THEOREM