VI

Contents

The Decomposition of Integer Polynomials 57

Eisenstein's Irreducibility Criterion 60

Chapter 7. The Construction of Regular Polygons 63

Constructions with Straightedge and Compass 69

The Classical Construction Problems 74

Chapter 8. The Solution of Equations of the Fifth Degree 81

The Transformations of Tschirnhaus and of Bring and

Jerrard 89

Chapter 9. The Galois Group of an Equation 93

Computing the Galois Group 114

A Quick Course in Calculating with Polynomials 119

Chapter 10. Algebraic Structures and Galois Theory 125

Groups and Fields 130

The Fundamental Theorem of Galois Theory: An

Example 144

Artin's Version of the Fundamental Theorem of

Galois Theory 149

The Unsolvability of the Classical Construction

Problems 161

Epilogue 165

Index 177