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,|
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