*Instructor:* Dr. 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 in number theory and abstract algebra as well
(requiring only the definition of rings, ideals and fields). It is strongly advised as a supplementary course
for those being interested in abstract algebra but lacking the basics.

*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, Horner arrangement

*multivariable polynomials: * symmetric polynomials, elementary
symmetric polynomials, the fundamental theorem of symmetric polynomials,
Newton formulae; Hilbert's basis theorem, Lüroth's theorem.

*polynomials over ***R*** and ***C**: the Fundamental
Theorem of Algebra, description of the irreducibles over **C ** and **R**, 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

*Polynomials over* **Z**_p : Exponentiating over **Z**_p, applications in number theory: Fermat's theorem, Wilson's theorem,
mod prime irreducibility test for integer polynomials

*Cyclotomic polynomials:* definition and calculation of
cyclotomic polynomials, $\Phi_n(x)\in {\bf Z[x]}$,
irreducibility of cyclotomic polynomials (no proof), roots of $\Phi_n(x)$
over $\bf{Z}_p$, special case of Dirichlet's Theorem: For every $n$ there are
infinitely many primes of the form $kn+1$.

**Permutations ** cycle decomposition, $S_n$ as a group, even and
odd permutations, conjugation among permutations

**Application of matrices and determinats** discriminant and
resultant, Vandermonde-determinants. Determinants of
block-matrices.

**Remark**.