CONTENT S 0 Background Section One Geometry 1 What Is Congruence? 2 Some Two-Dimensional Geometry 3 Symmetry Section Two Polynomials 4 The Root of It All 5 The Renaissance of Algebra 6 Complex Numbers 7 Symmetric Polynomials and The Fundamental Theorem of Algebra 8 Permutations and Lagrange's Theorem 9 Orbits and Cauchy's Formula 9A Hamilton's Quaternions (Optional) Section Three Numbers 10 Back to Euclid 11 Euclid's Lemma for Polynomials 12 Fermat and the Rebirth of Number Theory 13 Lagrange's Theorem Revisited 14 Rings and Squares 14A More Rings and More Squares 15 Fermat's Last Theorem (for Polynomials) 15A Still more Fermat's Last Theorem (Optional) xi 13 14 22 32 45 46 49 59 68 78 86 95 101 102 113 122 136 142 149 157 165

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