Study program
First cycle study programme in mathematics (Bachelor
level)
1st cycle
Study level
Course title
Algebra
Course code
MAT01-017
Language of instruction
English
Course objective. The objective of the course is to define
and study some basic algebraic structures.
Prerequisites. Geometry of plane and space. Linear
algebra I and II.
Course contents.
1. Groups. Groupoid, semigroup, monoid, group.
Homomorphisms and isomorphisms. Finite groups.
Lagrange's theorem. Normal subgroups and quotient
groups. Cyclic groups. Solvable groups. Sylow's
theorems.
Course description
2. Rings and modules. Rings. Examples. Multiplicative
group of a ring. Subring. Ideal. Quotient ring.
Homomorphisms and isomorphisms. Skew fields and
fields. Polynomial ring. Modules. Submodules and
quotient modules. Vector spaces.
3. Integral domains. Definition. Maximal ideals.
Characteristic. Simple fields. Fields of fractions.
Polynomial and rational functions.
4. Principal ideal rings. Definition. Finitely generated
modules over principal ideal rings. Classificaton of finite
Abelian groups. Connection with the theory of linear
operators.
5. Field extensions. Definition of field extensions. Finite
extensions. Degree of an extension. Algebraic extensions.
Transcendental elements of an extension. Purely transcendental extensions. Minimal polynomial. Simple
extensions. Algebraic closure.
6. Fundamental theorem of algebra. Sketch of a proof
using the notion of loops and their winding numbers.
7. Extensions of the field of rational numbers. Algebraic
and transcendental numbers. Gauss lemma. Eisenstein's
criterion of ireducibility of a polynomial. Gauss field and
Gauss integers. Algebraic integers. Quadratic extensions.
Constructibility by ruler and compass.
8. Galois theory. Splitting fields. Automorphisms of a
field. Galois group of a field exten-sion. Galois group of a
polynomial. Separable polynomials and separable
extensions. Basic theorems of Galois theory. Normal
extensions. Fundamental theorem of Galois theory.
9. Equations of third, fourth and higher degrees.
Cardano formulas. Solubility by radicals. Solutions of
a fourth degree equation. Fifth degree equation
insoluble by radicals.
Form of teaching
consultative teaching
Form of assessment
During the semester students can take tests which
replace the written part of the examination.
Number of ECTS
6
Class hours per week
2+2+0
Minimum number of
students
Period of realization
summer semester
Lecturer
Ivan Matić, Assistant Professor