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

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