Section 1: Geometry

• What Is Congruence?
• Some Two-Dimensional Geometry
• Symmetry

Section 2: Polynomials

• The Root of It All
• The Renaissance of Algebra
• Complex Numbers
• Symmetric Polynomials and The Fundamental Theorem of Algebra
• Permutations and Lagrange’s Theorem
• Orbits and Cauchy’s Formula
• Hamilton’s Quaternions (Optional)

Section 3: Numbers

• Back to Euclid; Euclid’s Lemma for Polynomials; Fermat and the Rebirth of Number Theory
• Lagrange’s Theorem Revisited; Rings and Squares
• More Rings and More Squares
• Fermat’s Last Theorem (for Polynomials)
• Still more Fermat’s Last Theorem (Optional)

Section 4: The Grand Synthesis

• Constructible Polygons and the Method of Mr. Gauss
• Cyclotomic Fields and Linear Algebra
• Lagrange Theorem for Fields and Nonconstructibility
• Galois Fields and the Fundamental Theorem of Algebra Revisited
• Galois’ Theory of Equations
• The Galois Correspondence
• Constructible Numbers and Solvable Equations