Text: handouts
Prerequisite: None.
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.
Topics:
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
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)