We will learn how the main ideas of modern algebra develop from two basic questions:

• Q:  How to solve equations?  A:  Ring theory.
• Q:  What types of symmetry are possible?   A:  Group theory.

Chapter references are to M. Artin, Algebra, which will be supplemented by handouts.

1. Number systems and rings
   1. Integers Z, Euclidean algorithm for gcd
      primes, unique factorization, irrationals
   2. Polynomial ring Q[x], polynomial division,factorization
      integer polynomials Z[x], Gauss Lemmas, Rational Root Test
   3. Ring axioms, Euclidean rings, field axioms, fraction fields
   4. Real numbers R, construction
      continuous functions, Intermediate Value Theorem
      complete ordered field axioms
   5. Complex numbers C, construction, geometric interpretation
      vector fields, analytic functions,line integrals, Cauchy Formulas
      Fundamental Theorem of Algebra

2. Groups and symmetry
   1. Group axioms, geometric examples
      homomorphisms, normal subgroups(Ch 2.1 -- 2.4)
   2. Plane symmetries, permutations (Ch 5.1 -- 5.4,   6.6)

3. Linear algebra and geometry
   1. Vector spaces, bases, coordinates (Ch 3.1 -- 3.4)
   2. Geometry in n dimensions
   3. Linear transformations (Ch 1.1,   4.1 -- 4.2)
   4. Determinants, eigenvectors, diagonalization(Ch 1.2 -- 1.3,   4.3,   4.6)