Instructor: Dr. Csaba Szabó

Text:  handouts

Prerequisite: None.

Course description:

This is a special non-credit three-week refresher crash course intented to teach/review basic notions and methods in classical algebra. The topics covered are needed e.g. in linear and abstract algebra. Thus it is especially advised as a supplementary course for those being interested in abstract algebra or Galois Theory, for example, but is strongly advised for everyone as the topics covered are useful in many other subjects.

As a general rule, if the sample problems below are mysterious you should consider (and are strongly advised) sitting in. More prcisely,
  • if you can solve 80% of this or at least two problems from this, you do not need to come to the CLA sessions of the first week (complex numbers).
  • If you can solve 50% of this, you do not need to come to the CLA sessions for the second and third weeks (polynomials).

Topics covered:

Week 1 — Complex Numbers
Introduction to complex numbers, algebraic and trigonometric forms, conjugation, length and norm, operations, n-th roots of a complex number, roots of unity, primitive roots of unity, the order of a complex number
geometric, algebraic and combinatorial applications of complex numbers

Weeks 2 and 3 — Polynomials
polynomials over fields: division algorithm, Euclidean algorithm, greatest common divisor, unique factorization of polynomials, polynomial functions
roots of polynomials: number of roots over fields, Viete-formulae -- the connection between the roots and the coefficients of the polynomial, multiple roots, formal differentiation, derivative-test,
multivariable polynomials: symmetric polynomials, elementary symmetric polynomials, the fundamental theorem of symmetric polynomials, Newton formulae;
polynomials over R and C: the Fundamental Theorem of Algebra, description of the irreducibles over R and C, algebraic closure.
Polynomials over Q and Z: integer and rational root tests, primitive polynomials, Gauss' lemma, Schoeneman-Eisenstein criteria for irreducibility, irreducible polynomials over the prime fields, Cyclotomic polynomials
Polynomials over Z_p: Exponentiating over Z_p, mod prime irreducibility test for integer polynomials