*Instructor: Csaba Szabó*

*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)