Instructor: Csaba Szabó
Course description:: The course provides basic notions and methods in classical algebra needed e.g. in linear and abstract algebra. However, it is intended to give a couple of concrete applications. It is strongly advised as a supplementary course for those being interested in abstract algebra but lacking the basics, but useful in many other subjects. As a general rule, if the sample problems in the syllabus below are mysterious, students should consider (and strongly advised) sitting in.
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
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
Remark If you can solve 80% of this or at least two problems from this, you do not need to come to the CLA session of the first week (complex numbers)
If you can solve 50% of this, you do not need to come to the CLA session for the second and third weeks (polynomials)