Language of Instruction

English

Level of Course Unit

First Cycle

Department / Program

MATHEMATICS

Mode of Delivery

Face to Face

Type of Course Unit

Elective

Objectives of the Course

The classical algebra problem is to find a ``formula'' for the roots of polynomial equations; by a formula, we mean a rule in terms of the coefficients of the polynomial obtained by just using addition, subtraction, multiplication, division and taking powers or roots (to any degree). Whether this is possible for any polynomial is the question that lead to Galois theory. Starting with this motivating problem, the aim is to develop all the necessary algebraic objects (groups and rings, polynomial rings, fields, field extensions) and their properties whenever needed on the way to solve that classical problem in field theory.

Course Content

Cubic and quartic equations. Cardan's Formulas. Symmetric polynomials. Discriminant. Roots of polynomials. The Fundamental Theorem of Algebra. Extension fields. Minimal polynomials. Adjoining elements. Degree of a field extension. Finite extensions. The tower theorem. Algebraic extensions. Simple extensions. Splitting fields, their uniqueness up to isomorphism. Normal extensions. Separable extensions. Fields of characteristic 0 and fields of characteristic p. The Primitive Element Theorem. Galois group. Galois group of splitting fields. Permutation of the roots. Examples of Galois groups. Abelian equations. Galois extensions. The Fundamental Theorem of Galois Theory. Solvability by radicals. Solvable groups. Cyclotomic extensions. Regular polygons and roots of unity. Impossibility of some geometric constructions using just straightedge and compass. Finite fields.

Course Methods and Techniques


Prerequisites and corequisities

None

Course Coordinator

None

Name of Lecturers

Prof.Dr. Dilek Pusat

Assistants

None

Work Placement(s)

No

Recommended or Required Reading
Resources

Redfield, R. H. Abstract Algebra, A Concrete Introduction, Pearson, 2001. Stewart, I. Galois Theory, Third edition, Chapman & Hall/CRC, 2003. Edwards, H. M. Galois Theory, Springer, 1984. Tignol, J. Galois' theory of algebraic equations, World Scientific, 2001.




Cox, David A. Galois Theory, WileyInterscience, 2004.





