Classical Algebra  CLA1

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.