*Instructor:*Dr. Csaba Szabó and Dr. Péter HERMANN

*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, |

### 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