MTHS 637
Topics in Algebra

James Reid

Course Schedule
Notes: These topics are meant to correspond, but only roughly, to class meetings. Pace and content may be modified at the discretion of the class.
Week 1 The Integers - The whole numbers, positive, negative and zero. Basic properties. Examples.
Week 2 The Relation of Divisibility and the Division Algorithm. Bases for the number system (Decimal, Binary, etc.). Examples.
Week 3 Primes, Greatest Common Divisors and Factorization. Irrational Numbers. Examples.
Week 4 Gauss and the Notion of Congruence. Fermat and Euler. Public Key Encryption. Examples.
Week 5 Polynomials - Basic properties. Examples.
Week 6 Divisibility and the Division Algorithm for Polynomials. Examples.
Week 7 Greatest Common Divisors. Factorization. The Remainder Theorem. Roots and Factors. Partial Fractions. Examples.
Week 8 Finding Roots. Creating Roots. The Methods of Hero and Newton. Cauchy. Solutions in Radicals. Examples.
Week 9 Algebraic Numbers. Transcendental Numbers. Notions of Galois Theory. A glimpse of algebraic number theory. Examples.
Week 10 Summary and Applications. Casting out nines. Decimal Representation of Fractions. Gaussian Integers. Pythagorean Triples.
Week 11 Some Geometry - Rigid Motions in the Plane. Similarities. Congruence and Similarity of geometric figures. Examples.
Week 12 The Group of Isometries. Its structure and the Classification of Isometries. Examples.
Week 13 Applications - The Conic Sections. Wallpaper Patterns. Foundations of Trigonometry.